All Order Merging of High Energy and Soft Collinear Resummation

Jeppe R. Andersen1, Hitham Hassan1$\star$ and Sebastian Jaskiewicz1

1 Institute for Particle Physics Phenomenology, Durham University, Durham,
United Kingdom

*<EMAIL_ADDRESS>
51st International Symposium on Multiparticle Dynamics (ISMD2022)
Pitlochry, Scottish Highlands, 1-5 August 2022 10.21468/SciPostPhysProc.?

## Abstract

We present a method of merging the exclusive LO-matched high energy
resummation of High Energy Jets (HEJ) with the parton shower of Pythia which
preserves the accuracy of the LO cross sections and the logarithmic accuracy
of both resummation schemes across all of phase space. Predictions produced
with this merging prescription are presented with comparisons to data from
experimental studies and suggestions are made for further observables and
experimental cuts which highlight the importance of both high energy and soft-
collinear effects.

##  1 Introduction

The emergence of complex divergent and slowly-convergent structures in
perturbation theory has significant implications on theorists’ ability to
produce stable and robust predictions for standard model processes. The direct
implication of these is that expansions in $\alpha_{s}$ converge slowly with
higher orders of expansion in certain regions of phase space as the size of
the logarithms compensates for the smallness of the coupling and an all-order
treatment is required. Most pertinent to these proceedings is the presence of
large logarithms to all orders of QCD in different kinematic limits.

In the high energy (HE) or multi-Regge kinematic (MRK) limit for a $2\to n$
scattering of partons with momenta $p_{a},p_{b}\to
p_{1},p_{2}\dots,p_{n-1},p_{n}$:

$y_{1}\gg y_{2}\gg\cdots\gg y_{n-1}\gg y_{n},\quad p_{i,\perp}\approx
k_{\perp}\quad\forall i\in\\{1,2,\cdots,n-1,n\\},$ (1)

with $k_{\perp}$ some hard transverse momentum scale, large logarithms
$\log(\hat{s}/k^{2}_{\perp})\to|\Delta y_{1,n}|$ arise to all orders in
perturbation theory [1]. Without an accounting for these effects, the phase
space corresponding to the MRK limit will be poorly modelled by fixed-order
predictions and descriptions of observables which receive this logarithmic
enhancement from semi-hard wide-angle radiation will not be perturbatively
stable.

These logarithms are most often resummed analytically via the Balitsky-Fadin-
Kuraev-Lipatov (BFKL) formalism [1, 2, 3, 4]. For this study we work instead
with the High Energy Jets [5] Monte Carlo implementation of high energy
resummation which matches the inclusive cross section to the leading order
(LO) prediction and applies the leading logarithmic\- (LL-)accurate high
energy corrections to processes which at tree level contribute at LL or NLL
accuracy [6].

The HEJ-resummable processes at LL (for pure dijet production) are scatterings
of the form $f_{1},f_{2}\rightarrow f_{1},g,\dots,g,f_{2}$ where the final
state is understood to be ordered in rapidity. We refer to such configurations
as “Fadin-Kuraev-Lipatov” or “FKL” configurations. The subleading
configurations which receive LL resummation in HEJ include those where the
rapidity ordering between any two neighbours in the final state is
relaxed111Sec 1.2.3 of ref [6] provides an overview of contributions from
subleading configurations and how these are resummed in HEJ.. Additional
subleading processes include those with $t$-channel quark propagators
(allowing for central $q\overline{q}$ emissions). The event output of HEJ is
exclusive to its logarithmic accuracy compared to the inclusive event input
generated at leading order.

In the limit of soft-collinear parton splittings a different class of
logarithms emerge which typically manifest as ratios of transverse energy
scales:

$\log^{2}\left(\frac{t_{j}}{t_{k}}\right),$ (2)

where $t$ has mass dimension +2. Such double logarithms may be recast into the
product of a soft logarithm and a collinear logarithm, each of which diverges
respectively when partons in an event have low transverse momenta or have
small angular separation. Just as with the high energy logarithms
$\log(\hat{s}/k^{2}_{\perp})$, the presence of these spoils the rapid
convergence of the perturbative series in the regions of phase space where
such logarithms are large. This includes, most plainly expressed in Eq. (2)
for a transverse momentum-based evolution scale, configurations with
hierarchies in $p_{\perp}$ and collinear splittings inside jet cones.

These effects may be accounted for with Monte Carlo parton showers which resum
the leading logarithmic soft-collinear effects to all orders in perturbation
theory. Most often inclusive fixed-order events are merged with parton shower
resummation. Merging events generated at fixed-order with parton showers is a
well-established field of research with many well-proven procedures including
CKKW-L [7, 8] merging for leading order input events. Methods for matching to
higher orders in perturbation theory, including MC@NLO [9] and the POWHEG [10]
methods for NLO events are also widely used. Further generalisations of these
methods for NLO matching which increase the accuracy below the merging scale
to full NLO have additionally been developed, including MENLOPS [11, 12] and
UNLOPS [13, 14]. The result of these are exclusive showered events with an
all-order description of the soft and collinear splittings.

We discuss here the implementation of a new procedure for merging the
exclusive high-energy-resummed event output of HEJ with the exclusive parton
shower resummation of Pythia8 [15] which accounts for both missing higher
order perturbative effects and systematically removes the double counted
contributions. The software implementation of this procedure is HEJ+Pythia.

##  2 Merging Procedure

To produce high energy- and soft-collinear-resummed predictions, we express
the resummation of HEJ in the language of the parton shower by defining a
splitting kernel corresponding to the HEJ matrix elements. In regular QCD, the
splitting functions may be calculated by: [16, 6]:

$dk_{\perp}^{2}dz\int
d\phi\frac{1}{16\pi^{2}}\frac{\left|\mathcal{M}^{n+1}\right|^{2}}{\left|\mathcal{M}^{n}\right|^{2}}\sim\frac{dk_{\perp}^{2}}{{\color[rgb]{0,0,1}k_{\perp}^{2}}}dz{\color[rgb]{0,0,1}\frac{\alpha_{s}}{2\pi}P(z)},$
(3)

with $\mathcal{M}$ the regular matrix elements of QCD. We may calculate the
analogue of the splitting function for HEJ analogously, substituting for the
regular matrix elements the HEJ-equivalent:

${\color[rgb]{0,0,1}P^{\text{\tt
HEJ}}}=\frac{1}{2}\frac{1}{16\pi^{2}}\frac{\overline{\left|\mathcal{M}_{\text{\tt
HEJ}}^{n+1}\right|^{2}}}{\overline{\left|\mathcal{M}_{\text{\tt
HEJ}}^{n}\right|^{2}}},$ (4)

where the extra factor of $1/2$ in the HEJ splitting probability arises when
we consider colour configurations. There are two possible colour
configurations for inserting a particle into a HE configuration [17, 18] which
we weight equally in our treatment. The quantities highlighted in blue from
Eq. (3) are implicitly contained in $P^{\text{\tt HEJ}}$. Our method borrows
from CKKWL merging in a similar manner to ref. [19] by introducing a scheme
within which shower emissions are vetoed according to the probability they
have already been accounted for by HEJ:

$\mathcal{P}^{\mathrm{veto}}=\frac{P^{\textrm{\tt HEJ}}}{P^{\textrm{\sc
Pythia}}}\cdot\Theta(P^{\textrm{\sc Pythia}}-P^{\textrm{\tt
HEJ}})+1\cdot\Theta(-P^{\textrm{\sc Pythia}}+P^{\textrm{\tt HEJ}}),$ (5)

with $P^{\text{\sc Pythia}}$ the Altarelli-Parisi splitting kernels of QCD
(weighted by $\alpha_{s}/2\pi k_{\perp}^{2}$). For the case of an initial
state emission $i\to jk$, the splitting kernel must be reweighted by the
correct ratio of parton distribution functions (PDFs) to properly reproduce
the backwards DGLAP evolution:

$P\rightarrow
P\cdot\frac{x_{i}f_{i}(x_{i},\mu_{F}^{2})}{x_{j}f_{j}(x_{j},\mu_{F}^{2})},$
(6)

with $f_{i,j}$ the relevant PDF at energy fraction $x_{i,j}$ and an
appropriately chosen factorisation scale $\mu_{F}$. The combined effect of the
above is that for emissions with $P^{\text{\sc Pythia}}<P^{\textrm{\tt HEJ}}$
we veto trial emissions with 100% probability and thus the shower emissions
are produced with a modified Sudakov form factor:

$\Delta^{S}(k^{2}_{\perp,i},k^{2}_{\perp,i+1})=\exp\left\\{-\int^{k^{2}_{\perp,i}}_{k^{2}_{\perp,i+1}}dk^{2}_{\perp}dz\,\Theta\left(P^{\textrm{\sc
Pythia}}-P^{\textrm{\tt HEJ}}\right)\left[P^{\textrm{\sc
Pythia}}(k^{2}_{\perp},z)-P^{\textrm{\tt
HEJ}}(k^{2}_{\perp},z)\right]\right\\},$ (7)

which subtracts from the Pythia splitting kernel the equivalent for HEJ.

For HEJ-resummable input events, we construct histories \- sequences of
splittings that connect the LO $2\to 2$ scattering (for the case of inclusive
dijet production) to the $2\to n$ input event, ordered in the Pythia shower
evolution variable $k_{\perp}^{2}$. We assign to each history a weight
proportional to the product of HEJ splitting probabilities and select one
according to their weights. Our procedure for merging such configurations may
be summarised as follows:

  1. 1.

Start the trial shower with an emission from state $i$ in the history with
scale $k^{2}_{\perp,i}$ (starting with $i=0$).

    1. (a)

if $k^{2}_{\perp,i+1}<k^{2}_{\perp,i}$, continue to the next state in the
history, setting $i\rightarrow i+1$ and returning to step 1. If this is the
original event, go to step 2.

    2. (b)

if $k^{2}_{\perp,i+1}>k^{2}_{\perp,i}$, veto the emission with probability
$\mathcal{P}^{\mathrm{veto}}$ of Eq. (5).

      1. i.

If vetoed generate a new trial emission from the current scale and return to
step 1a.

      2. ii.

If not vetoed, keep the trial emission and append it to each subsequent node
in the history $i+1,i+2,...$ . Generate a new trial emission from the current
scale and return to step 1a.

  2. 2.

Perform a final trial emission (to account for the case no emissions have yet
been appended due to the veto) and exit the merging after accepting an
emission according to the veto probability in 1b, using the updated event
record to initiate the later shower.

  3. 3.

Continue the parton shower evolution on the merged event (i.e. the input HEJ
event dressed with the additional shower emissions at all stages of the
history), veto each trial emission with probability
$\mathcal{P}^{\mathrm{veto}}$ until the hadronisation scale is reached.

  4. 4.

Hadronise the event.

We note several developments since the previous merging procedure of ref. [19]
(which itself was a development of ref. [20]). Most noticeably, all of the
original HEJ partons from the input event are retained (up to later
splittings) in our merging procedure meaning that the coverage of phase space
probed during high energy resummation is conserved. In the previous
implementation the shower phase space was unrestricted, but the merging
procedure would terminate after accepting one emission from Pythia thus not
guaranteeing the retention of high energy or shower logarithmic accuracy
beyond the first perturbative correction.

Secondly, much development in the HEJ formalism has allowed for a greater
number of configurations to be resummed including those contributing to
subleading logarithmic accuracies in BFKL [6]. This means the definition of
HEJ-states as they are referred to in ref. [19] has been widened to include
configurations with a central/extremal (in rapidity) $q\overline{q}$ pair or
an extremal gluon at LO as we have discussed.

Thirdly, the non-HEJ-resummable states at LO are merged via ordinary CKKWL in
Pythia as this procedure has been designed to account for the double counting
of emissions. We thus retain the inclusive cross section and implement a
completely unitary procedure which perserves the logarithmic accuracy of both
resummation schemes to all orders in perturbation theory across phase space.

##  3 Results

We present here validation predictions for the HEJ+Pythia merging procedure by
comparing our results to experimental data from the ATLAS collaboration of jet
profiles for inclusive jet production at $\sqrt{s}=7$ TeV [21]. Jets were
clustered with a radius parameter $R=0.6$ and a minimum transverse momentum of
$p_{\perp,j}>30$ GeV, with the additional requirement that they were central
in rapidity i.e. $|y_{j}|<2.8$. The differential jet profile $\rho(r)$ is
defined by:

$\rho(r)=\frac{1}{\Delta
r}\frac{1}{N_{\mathrm{jets}}}\sum_{\mathrm{jets}}\frac{p_{T}(r-\Delta
r/2,r+\Delta r/2)}{p_{T}(0,R)},$ (8)

with $R$ the jet radius parameter and $p_{T}(r_{1},r_{2})$ the (scalar) sum of
the transverse momentum in an annulus (in $y$-$\phi$ space) between radii
$r_{1}$ and $r_{2}$. The integrated jet profile $\Psi(r)$ is defined as the
normalised integral of $\rho$ between 0 and $r$.

The region of phase space probed by such measurements receives strong soft and
collinear enhancement and small corrections from BFKL effects. Indeed, HEJ
predicts jet profiles overwhelmingly dominated by one central hard parton as
demonstrated in Fig. 1. Parton showers excel in their description of such
observables as they resum the very effects required to populate the regions of
phase space probed - thus we expect little difference between a pure Pythia
prediction and our merged HEJ+Pythia prediction.

Figure 1: Comparisons of HEJ, HEJ+Pythia and Pythia to experimental data for
differential jet profiles (left) and integrated jet profiles (right) with a
hard jet selection. The showered predictions have MPI and hadronisation
enabled with “Monash 2013" the chosen tune. Data from ATLAS [21] \- the cuts
of this experimental analysis are displayed on the figure.

This is indeed the trend shown in Fig. 1 with both showered predictions
yielding robust descriptions of the ATLAS data [21]. This is a reassuring
validation test of our algorithm and demonstrates that we are able to dress
HEJ-resummed states with the missing higher order shower logarithms.

##  4 Conclusion

The first validation tests of our merging procedure which remains in
development are highly encouraging. This study has proved further that the
intersection of different resummation schemes is rarely straightforward and
treating the two regimes discussed in this study as “opposites" does not lend
itself to producing physical descriptions of standard model processes [19].

Our aims for this work are to extend the formalism to include $H$, $Z/\gamma$,
and $W$ production with jets such that all processes treated by HEJ may be
produced with physical shower evolution — bringing the hard partonic
predictions of HEJ closer to experimental truth-level. We additionally seek to
produce predictions for distributed cross sections and ratios (e.g. $R_{32}$)
in terms of the pertinent experimental observables to these resummation
schemes. These include (and are not limited to) $p_{\perp}$-based observables
for the shower predictions and dijet invariant masses and rapidity differences
to test the preservation of the HEJ logarithmic accuracy.

The types of experimental analysis which would probe regions of phase space
with high energy and soft-collinear enhancement would include an inclusive set
of cuts such as the following:

  1. 1.

Wide rapidity selection e.g. $|y|<4.5$ rather than restricting to the central
region $|y|<2.8$ — this will test regions where HEJ corrections are applied.

  2. 2.

Hard jets with $p_{\perp}$-hierarchy in jet selection e.g. $p_{\perp,j}>60$
GeV, with $p_{\perp,j_{1}}>80,100,120$ GeV — this will test the correct
application of shower corrections, varying this gives an indication of the
size of the effect.

  3. 3.

Jets of varying radii ($R=0.4,\,0.6$) — this will explore the effect of
broadening the jet cone on each resummation scheme.

This will demonstrate that neither resummation on its own will stably model
the entirety of the available phase space and explore the method we have
developed for properly accounting for both missing higher order effects.

## Acknowledgements

We would like to thank the organising committe for inviting us to present our
research and for organising a most productive and engaging conference. We also
thank our colleagues in the HEJ collaboration for our discussions during this
project. HH gratefully acknowledges funding by the UK STFC under grant
ST/T506047/1.

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# Invariance of Schur multiplier, Bogomolov multiplier and the minimal number
of generators under a variant of isoclinism

A. E. Antony<EMAIL_ADDRESS>Sathasivam K<EMAIL_ADDRESS>V.Z. Thomas  School of Mathematics, Indian Institute of Science Education and
Research Thiruvananthapuram,
695551 Kerala, India<EMAIL_ADDRESS>
###### Abstract

We introduce the $q$-Bogomolov multiplier as a generalization of the Bogomolov
multiplier, and we prove that it is invariant under $q$-isoclinism. We prove
that the $q$-Schur Multiplier is invariant under $q$\- exterior isoclinism,
and as an easy consequence we prove that the Schur multiplier is invariant
under exterior isoclinism. We also prove that if $G$, $H$ are $p$-groups and
$G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$, then the cardinality of the minimal
number of generators of $G$ and $H$ are the same. Moreover we prove some
structural results about $q$-nonabelian tensor square of groups.

###### keywords:

Schur Multiplier , Bogomolov Multiplier , isoclinism , group actions.

###### MSC:

[2020] 20J05 , 20J06 , 19C09 , 20D15 , 20F05 , 20F14 , 20F18

##  1 Introduction

The notion of isoclinism was introduced by P. Hall in [10]. It is a well-known
fact that the Schur Multiplier of any group and the minimal number of
generators of a $p$-group are not invariant under isoclinism. In this paper,
we define a variant of isoclinism and prove that the Schur Multiplier and the
bogomolov multiplier are invariant under this modified version of isoclinism.
Invariance of the Bogomolov Multiplier under isoclinism was proved by
Bogomolov and Böhning in [3] and by Moravec in [12]. We define two different
notions of $q$-isoclinism and consider two variants of the Bogomolov
Multiplier, and we further prove their invariance under $q$-isoclinism. We
also prove that if $G$, $H$ are $p$-groups and $G/Z^{\wedge}(G)\cong
H/Z^{\wedge}(H)$, then $d(G)=d(H)$, where $d(G)$ is the minimal number of
generators of $G$. For a group $G$ and a positive integer $q$, the nonabelian
$q$-tensor product and nonabelian $q$-exterior product for $G$-crossed modules
were introduced in [7], as a generalization of definitions given in [4] and
[8]. In this paper, we consider a special case of this where the crossed
modules are given by a group $G$ and the identity morphism $Id:G\rightarrow
G$, called the nonabelian q-tensor square of $G$. In [1], the authors prove
several structural results for the case $q=0$. We generalize these structural
results for a general $q$ and generalize some result of [13].

##  2 Preliminary Results

For $q\geq 1$, the $q$-nonabelian tensor square is defined as follows.

######  Definition 2.1.

The tensor square modulo q, $G\otimes^{q}G$ of the $G$-crossed module
$Id:G\rightarrow G$ is the group generated by the symbols $g\otimes h$ and
$\\{(g,g)\\},g,h\in G$ with the following relations for $g,h,g_{1},h_{1}\in
G:$

$gg_{1}\otimes h=(^{g}g_{1}\otimes\ ^{g}h)(g\otimes h),$ (1) $g\otimes
hh_{1}=(g\otimes h)(^{h}g\otimes\ ^{h}h_{1}),$ (2) $\\{(g,g)\\}(g_{1}\otimes
h_{1})\\{(g,g)\\}^{-1}=\hskip 5.69054pt^{g^{q}}g_{1}\otimes\hskip
1.42262pt^{g{{}^{q}}}h_{1},$ (3)
$\\{(gg_{1},gg_{1})\\}=\\{(g,g)\\}\prod_{i=1}^{q-1}(g^{-1}\otimes(^{g^{1-q+i}}g_{1})^{i})\\{(g_{1},g_{1})\\},$
(4) $[\\{(g,g)\\},\\{(g_{1},g_{1})\\}]=g^{q}\otimes{g_{1}}^{q},$ (5)
$\\{([g,h],[g,h])\\}=(g\otimes h)^{q}.$ (6)

For $q=0$, the $q$-tensor square is defined to be the construction introduced
by R. Brown and J.-L. Loday in [5] and [6], called the nonabelian tensor
square of the group $G$. Let $\nabla^{q}(G)$ denote the normal subgroup in
$G\otimes^{q}G$ generated by elements of the form $g\otimes^{q}g$, where $g\in
G$. The exterior square modulo q, $G\wedge^{q}G$, is the quotient of the group
$G\otimes^{q}G$ by $\nabla^{q}(G)$. The coset represented by the element
$g\otimes^{q}h$ is denoted by $g\wedge^{q}h$. Let $\Delta^{q}(G)$ be the
normal subgroup in $G\otimes^{q}G$ generated by elements of the form
$(g\otimes^{q}h)(h\otimes^{q}g)$, where $g,h\in G$.

Now we collect some results that we will need in later sections.

######  Lemma 2.2.

Let $G$ be a group. For $g,g_{1},h,h_{1}\in G$ and $n\in Z$, the following
hold in $G\otimes^{q}G$.

  * (i)

$\nabla^{q}(G)$ and $\Delta^{q}(G)$ commute with image of $G\otimes G$ in
$G\otimes^{q}G$ under the natural map.

$\displaystyle[g\otimes g,a\otimes b]=1,$ $\displaystyle[(g\otimes h)(h\otimes
g),a\otimes b]=1,$

for all $g,h,a,b\in G$. In particular, both $\nabla^{q}(G)$ and
$\Delta^{q}(G)$ are abelian.

  * (ii)

$\exp(\nabla^{q}(G))\mid q$ for $q>0$.

  * (iii)

$G$ acts trivially on $\nabla^{q}(G)$. In particular, $G$ acts trivially on
$\Delta^{q}(G)$.

  * (iv)

$[g\otimes h,g_{1}\otimes h_{1}]=[g,h]\otimes[g_{1},h_{1}].$

  * (v)

$((g^{-1}\otimes g_{1})(g_{1}\otimes g^{-1}))^{-1}=(g_{1}\otimes g)(g\otimes
g_{1}).$

  * (vi)

$(gg_{1}\otimes gg_{1})=(g\otimes g)(g_{1}\otimes g)(g\otimes
g_{1})(g_{1}\otimes g_{1}).$ In particular,
$\Delta^{q}(G)\subseteq\nabla^{q}(G)$.

  * (vii)

If $[g,h]=1$, then $(g\otimes h^{n})=(g\otimes h)^{n}=(g^{n}\otimes h).$

  * (viii)

$(g\otimes h)(g_{1}\otimes h_{1})(g\otimes h)^{-1}=\ ^{[g,h]}(g_{1}\otimes\
h_{1}).$

  * (ix)

If $q$ is odd and $[g,h]=1$, then $\\{(gh,gh)\\}=\\{(g,g)\\}\\{(h,h)\\}.$

  * (x)

$(g\otimes h^{n})(h^{n}\otimes g)=\big{(}(g\otimes h)(h\otimes
g)\big{)}^{n}=(g\otimes h)^{n}(h\otimes g)^{n}$

  * (xi)

If $x\in[G,G]$, then $x\otimes x=1$ and $(x\otimes g)(g\otimes x)=1$ for every
$g\in G$.

  * (xii)

$g^{-1}\otimes g=(g\otimes g)^{-1}=g\otimes g^{-1}$ and $g^{-1}\otimes
g^{-1}=g\otimes g.$

  * (xiii)

$\\{(1,1)\\}=1.$

##  3 Invariance of Schur Multiplier and the minimal number of generators

The notion of isoclinism was introduced by P. Hall in [11]. Two groups $G$ and
$H$ are said to be isoclinic if there exists isomorphisms
$\alpha:G/Z(G)\rightarrow H/Z(H)$ and $\beta:G^{\prime}\rightarrow H^{\prime}$
such that the following diagram commutes:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
49.04431pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G/Z(G)\times
G/Z(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
61.82887pt\raise 5.33333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\alpha\times\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 93.04431pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-29.59721pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-47.80556pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
93.04431pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{H/Z(H)\times
H/Z(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
144.05803pt\raise-29.59721pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
144.05803pt\raise-47.80556pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.04932pt\raise-53.08333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
135.72556pt\raise-59.19443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
135.72556pt\raise-59.19443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{\prime}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $\pi_{1}$ and $\pi_{2}$ are the commutator maps.

It is well-known that the Schur multiplier and the minimal number of
generators of a $p$-group is not invariant under the notion of isoclinism. In
this section, we define a variant of the notion of isoclinism under which the
Schur multiplier and the minimal number of generators of a $p$-group become
invariant.

Consider the homomorphism $\eta:G\wedge^{q}G\rightarrow G$ defined by $x\wedge
y\mapsto[x,y]$ and $\\{(g,g)\\}\mapsto g^{q}$ in [7]. The kernel of this
homomorphism is called the $q$-Schur Multiplier of $G$, and is denoted by
$M^{q}(G)$. For $q\geq 1$, we have the following normal subgroups of $G$:

$\displaystyle Z^{\wedge}(G)$ $\displaystyle=\\{g\in G\mid g\wedge h=1,\
\forall h\in G\\},$ $\displaystyle Z^{\wedge}_{q}(G)$ $\displaystyle=\\{g\in
G\mid g\wedge^{q}h=1,\ \forall h\in G\\},$ $\displaystyle E^{\wedge}_{q}(G)$
$\displaystyle=\\{g\in Z^{\wedge}_{q}(G)\mid\\{(g,g)\\}=1\in G\wedge^{q}G\\}.$

For $q=0$, set $E^{\wedge}_{0}(G):=Z^{\wedge}(G)$.

######  Definition 3.1.

We define two groups $G$ and $H$ to be $q$-exterior isoclinic if there exists
a pair of isomorphisms $\alpha:G/E_{q}^{\wedge}(G)\rightarrow
H/E_{q}^{\wedge}(H)$ and $\beta:G^{q}[G,G]\rightarrow H^{q}[H,H]$ with the
property that whenever $\alpha(g_{1}E_{q}^{\wedge}(G))=h_{1}E_{q}^{\wedge}(H)$
and $\alpha(g_{2}E_{q}^{\wedge}(G))=h_{2}E_{q}^{\wedge}(H),$ then
$\beta([g_{1},g_{2}])=[h_{1},h_{2}]$ and $\beta(g_{1}^{q})=h_{1}^{q}$ for
$g_{1},g_{2}\in G$. When $q=0$, we will call $0$-exterior isoclinism as
exterior isoclinism.

The exterior isoclinism is an equivalence relation and all cyclic groups are
exterior isoclinic to {1}.

######  Definition 3.2.

We define two groups $G$ and $H$ to be weak $q$-exterior isoclinic if there
exists subgroups $A\leqslant E^{\wedge}_{q}(G)$, $B\leqslant
E^{\wedge}_{q}(G)$ and a pair of isomorphism $\alpha:G/A\rightarrow H/B$,
$\beta:G^{q}[G,G]\rightarrow H^{q}[H,H]$ with the property that whenever
$\alpha(g_{1}A)=h_{1}B$ and $\alpha(g_{2}A)=h_{2}B,$ then
$\beta([g_{1},g_{2}])=[h_{1},h_{2}]$ and $\beta(g_{1}^{q})=h_{1}^{q}$ for
$g_{1},g_{2}\in G$

######  Theorem 3.3 (Invariance of $q$-Schur multiplier).

Let $G$ and $H$ be two groups, and $A\leqslant E^{\wedge}_{q}(G)$, $B\leqslant
E^{\wedge}_{q}(H)$.

  1. (i)

If $G/A{\cong}H/B$, then $G\wedge^{q}G\cong H\wedge^{q}H.$

  2. (ii)

If $G$ and $H$ are weak $q$-exterior isoclinic then $M^{q}(G)\cong M^{q}(H)$.
In particular, $M^{q}(G)$ is invariant under $q$-exterior isoclinism.

###### Proof.

(i) Consider the exact sequence $A\wedge^{q}G\rightarrow
G\wedge^{q}G\rightarrow G/A\wedge^{q}G/A\rightarrow 1$ found in [9,
Proposition 7]. Noting that the image of $A\wedge^{q}G$ is trivial in
$G\wedge^{q}G$ yields $G\wedge^{q}G\cong G/A\wedge^{q}G/A$. The isomorphism
$\phi:G/A{\cong}H/B$ induces an isomorphism $\bar{\phi}:G/A\wedge^{q}G/A\to
H/B\wedge^{q}H/B.$ Hence we obtain, $G\wedge^{q}G\cong G/A\wedge^{q}G/A\cong
H/B\wedge^{q}H/B\cong H\wedge^{q}H.$

(ii) Consider the following commutative diagram :

${1}$${M^{q}(G)}$${G\wedge^{q}G}$${G^{q}{[G,G]}}$${1}$${1}$${M^{q}(H)}$${H\wedge^{q}H}$${H^{q}{[H,H]}}$${1.}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\beta}$

Using (i), $\bar{\phi}$ is isomorphism. Now the short five lemma yields the
result. ∎

######  Corollary 3.4 (Invariance of Schur Multiplier under exterior
isoclinism).

If $G$ and $H$ are two groups that are exterior isoclinic, then $M(G)\cong
M(H)$.

######  Corollary 3.5.

Let $G$ be a group, and $H\leqslant G$, $K\leqslant Z^{\wedge}(G).$ If $G=HK$
and $H\cap K\leqslant Z^{\wedge}(H)$, then $M(G)\cong M(H)$.

###### Proof.

$G/K=HK/K\cong H/(H\cap K)$ and the inclusion $i:[H,H]\to[G,G]$ is an
isomorphism. It is easy to verify $G$ and $H$ are weak exterior isoclinic and
hence the result follows from theorem 3.3 (ii). ∎

######  Definition 3.6.

We say $G$ is internal central product of $H$ and $K$ amalgamating $D$ if
$G=HK$, $H\cap K=D$ and $[H,K]=1$

######  Corollary 3.7.

Let $G$ be a internal central product of $H$ and $K$ amalgamating $D$. If $K$
cyclic, $(|H/D|,|K/D|)=1$ and $D\leqslant Z^{\wedge}(H)$, then $M(G)\cong
M(H).$

###### Proof.

It suffices to show $K\leqslant Z^{\wedge}(G)$. Consider the morphisms
$H\wedge H\to G\wedge G$. Since $D\leqslant Z^{\wedge}(H)$, we have $h\wedge
d=1$ in $H\wedge H$ and hence $h\wedge d=1$ in $G\wedge G$, for $h\in H$,
$d\in D$. Similarly $k\wedge k_{1}=1$ in $G\wedge G$, where $k,k_{1}\in K$.
Denote $|H/D|$ by $n$ and let $kD$ be a generator of $K/D$. Since
$(|H/D|,|K/D|)=1$, $k^{n}D$ is also a generator of $K/D$. Thus an element of
$K$ is of the form $k^{ni}d$ for some $d\in D,i\in\mathbb{Z}$. For any $h\in
H$,

$\displaystyle k^{ni}d\wedge h$ $\displaystyle=\;^{k^{ni}}(d\wedge
h)(k^{ni}\wedge h)$ $\displaystyle=(k^{ni}\wedge h)=(k\wedge h)^{ni}=k\wedge
h^{ni}$ $\displaystyle\mbox{[Lemma}\ \ref{L:1.2.3}\ \textit{(vii)}]$
$\displaystyle=k\wedge d_{1}=1,\text{ for some $d_{1}\in D.$}$

Any $g\in G$ is of the form $k_{1}h$ for some $k_{1}\in K$ and $h\in H$.
Therefore, $k^{ni}d\wedge g=k^{ni}d\wedge k_{1}h=1$, and hence $K\leqslant
Z^{\wedge}(G)$. ∎

######  Corollary 3.8.

Let $K$ be a cyclic group, and $G=H\times K$. If order of $H$ is co-prime to
the order of $K$, then $M(G)\cong M(H).$

The next lemma shows the universal property of the quotient group
$G/{Z^{\wedge}(G)}$. It says that a surjective group morphism from any group
$G$ to a capable group factors through $G/{Z^{\wedge}(G)}$.

######  Lemma 3.9.

Let $G$ be a group and $N$ be a normal subgroup of $G$. If $G/N$ is capable,
then $Z^{\wedge}(G)\leqslant N.$

###### Proof.

Let $\pi:G\to G/N$ be the projection map. We will show that if $g\in
G\setminus N$, then $g\notin Z^{\wedge}(G)$. Let $g\in G\setminus N$, we have
$\pi(g)\neq 1$. Since $G/N$ is capable, $\pi(g)\notin Z^{\wedge}(G/N)$. Thus,
there exists $\bar{h}\in G/N$ such that $\pi(g)\wedge\bar{h}\neq 1.$ Let $h\in
G$ with $\pi(h)=\bar{h}$, and note that $g\wedge h\neq 1$. So $g\notin
Z^{\wedge}(G)$. ∎

######  Corollary 3.10.

For a finite non-cyclic $p$-group $G$, we have
$Z^{\wedge}(G)\leqslant\Phi(G).$

###### Proof.

$G/\Phi(G)$ is a non-cyclic elementary abelian group. Thus $G/\Phi(G)$ is
capable by [2] and the result follows by lemma 3.9. ∎

######  Corollary 3.11.

Let $G$ be a finite non-cyclic group. If
$G/[G,G]\cong\mathbb{Z}_{n_{1}}\times\cdots\times\mathbb{Z}_{n_{k}}\times\mathbb{Z}_{n_{k+1}}$,
$n_{i}|n_{i+1}$ for $1\leq i\leq k$, then
$Z^{\wedge}(G)\leqslant[G,G]G^{n_{k}}.$ In particular, if $G/[G,G]$ is
capable, then $Z^{\wedge}(G)\leqslant[G,G]$

###### Proof.

Observe that
$G/[G,G]G^{n_{k}}\cong\mathbb{Z}_{n_{1}}\times\cdots\times\mathbb{Z}_{n_{k}}\times\mathbb{Z}_{n_{k}}$.
Thus $G/[G,G]G^{n_{k}}$ is capable by [2] and the result follows by lemma 3.9.
∎

Recall that a unicentral group is a group whose exterior center coincides with
its center. Denoting the minimal number of generators of $G$ by $d(G)$, we
have the following theorem:

######  Theorem 3.12.

Let $G$ and $H$ be finite $p$-groups. If $G/Z^{\wedge}(G)\cong
H/Z^{\wedge}(H)$, then $d(G)=d(H)$. In particular, if $G$ and $H$ are exterior
isoclinic, then $d(G)=d(H)$.

###### Proof.

Suppose $d(G)=1$, then $G=Z^{\wedge}(G).$ Since $G/Z^{\wedge}(G)\cong
H/Z^{\wedge}(H)$, we have $H=Z^{\wedge}(H).$ Thus $H$ is a unicentral group.
We claim that a unicentral finite abelian group is cyclic. Suppose not, then
$H\cong\mathbb{Z}_{n_{1}}\times\cdots\times\mathbb{Z}_{n_{k}}\times\mathbb{Z}_{n_{k+1}}$,
and
$H/H^{n_{k}}\cong\mathbb{Z}_{n_{1}}\times\cdots\times\mathbb{Z}_{n_{k}}\times\mathbb{Z}_{n_{k}}$,
and hence $H/H^{n_{k}}$ is capable. By Proposition 3.9, we obtain that
$Z^{\wedge}(H)\leqslant H^{n_{k}}$, a contradiction. Thus $H$ is cyclic and so
$d(H)=1$. We may now assume that $d(G)\geq 2$ and $d(H)\geq 2$. By corollary
3.10, $Z^{\wedge}(G)\leqslant\Phi(G)$ and so
$\Phi(G/Z^{\wedge}(G))=\Phi(G)/Z^{\wedge}(G)$. Thus, we have
$\frac{G/Z^{\wedge}(G)}{\Phi(G/Z^{\wedge}(G))}=\frac{G/Z^{\wedge}(G)}{\Phi(G)/Z^{\wedge}(G)}\cong
G/\Phi(G).$ Hence,
$d(G)=dim_{\mathbb{F}_{p}}(G/\Phi(G))=dim_{\mathbb{F}_{p}}(\frac{G/Z^{\wedge}(G)}{\Phi(G/Z^{\wedge}(G))})=d(G/Z^{\wedge}(G))$.
Similarly $d(H)=d(H/Z^{\wedge}(H))$. Moreover the isomorphism
$G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$ yields
$d(G/Z^{\wedge}(G))=d(H/Z^{\wedge}(H))$, and hence $d(G)=d(H)$. ∎

##  4 Invariance of q-Bogomolov Multiplier

In this section, we consider two distinct quotients of the $q$-Schur
Multiplier of groups. Two notions of isoclinism are defined and it is observed
that the considered quotients are invariant under the appropriate notion of
isoclinism. The $q$-Bogomolov Multiplier, $B_{0}^{q}(G)$, and
$\widehat{q}$-Bogomolov Multiplier , $\widehat{B_{0}^{q}(G)}$, are defined
using the normal subgroups $M_{0}^{q}(G)$ and $\widehat{M_{0}^{q}(G)}$ of the
$q$-Schur Multiplier described below.

$\displaystyle M_{0}^{q}(G)$ $\displaystyle:=\left\langle x\wedge
y|[x,y]=1\right\rangle.$ $\displaystyle B_{0}^{q}(G)$
$\displaystyle:=M^{q}(G)/M_{0}^{q}(G).$ $\displaystyle\widehat{M_{0}^{q}(G)}$
$\displaystyle:=\left\langle x\wedge
y,\\{(g,g)\\}|[x,y]=1,g^{q}=1\right\rangle.$
$\displaystyle\widehat{B_{0}^{q}(G)}$
$\displaystyle:=M^{q}(G)/\widehat{M_{0}^{q}(G)}.$

We have following normal subgroups of $G$.

$\displaystyle Z_{q}(G)$ $\displaystyle:=\\{g\in Z(G)\mid\\{(g,g)\\}\in
M_{0}^{q}(G)\\}.$ $\displaystyle\widehat{Z_{q}(G)}$ $\displaystyle:=\\{g\in
Z(G)\mid g^{q}=1\\}.$

######  Definition 4.1.

Two groups $G$ and $H$ are $q$-isoclinic if there exists isomorphisms
$\alpha:G/Z_{q}(G)\rightarrow H/Z_{q}(H)$ and $\beta:G^{q}[G,G]\rightarrow
H^{q}[H,H]$ with the property that whenever
$\alpha(a_{1}Z_{q}(G))=a_{2}Z_{q}(H)$ and
$\alpha(b_{1}Z_{q}(G))=b_{2}Z_{q}(H),$ then
$\beta([a_{1},b_{1}])=[a_{2},b_{2}]$ and $\beta(a_{1}^{q})=a_{2}^{q}$ for
$a_{1},b_{1}\in G$, for $a_{1},b_{1}\in G$.

######  Definition 4.2.

Two groups $G$ and $H$ are $\widehat{q}$-isoclinic if there exists
isomorphisms $\alpha:G/\widehat{Z_{q}(G)}\rightarrow H/\widehat{Z_{q}(H)}$ and
$\beta:G^{q}[G,G]\rightarrow H^{q}[H,H]$ with the property that whenever
$\alpha(a_{1}\widehat{Z_{q}(G)})=a_{2}\widehat{Z_{q}(H)}$ and
$\alpha(b_{1}\widehat{Z_{q}(G)})=b_{2}\widehat{Z_{q}(H)},$ then
$\beta([a_{1},b_{1}])=[a_{2},b_{2}]$ and $\beta(a_{1}^{q})=a_{2}^{q}$ for
$a_{1},b_{1}\in G$.

######  Theorem 4.3.

$B_{0}^{q}(G)$ and $\widehat{B_{0}^{q}(G)}$ are invariant under $q$-isoclinism
and $\widehat{q}$-isoclinism respectively.

###### Proof.

Given isomorphisms $\alpha$ and $\beta$ as in the definition of
$q$-isoclinism, we first show that the following map is a well defined
isomorphism:

$\displaystyle\phi:\hskip 4.2679pt$ $\displaystyle
G\wedge^{q}G/M_{0}^{q}(G)\rightarrow H\wedge^{q}H/M_{0}^{q}(H)$
$\displaystyle\overline{x_{1}\wedge y_{1}}\mapsto\overline{x_{2}\wedge
y_{2}},\ \overline{\\{(x_{1},x_{1})\\}}\mapsto\overline{\\{(x_{2},x_{2})\\}},$

where $\alpha(x_{1}Z_{q}(G))=x_{2}Z_{q}(H)$ and
$\alpha(y_{1}Z_{q}(G))=y_{2}Z_{q}(H)$. Towards that, consider the map
$\phi:G\wedge^{q}G\rightarrow\hskip 2.84526ptH\wedge^{q}H/M_{0}^{q}(H)\hskip
2.84526pt$ with $x_{1}\wedge y_{1}\mapsto\overline{x_{2}\wedge y_{2}}$ and
$\\{(x_{1},x_{1})\\}\mapsto\overline{\\{(x_{2},x_{2})\\}}$, where $x_{2}$ and
$y_{2}$ are obtained as above. For
$\alpha(\overline{x_{1}})=\overline{x_{2}a_{2}}$ and
$\alpha(\overline{y_{1}})=\overline{y_{2}b_{2}}$, where $a_{2},b_{2}\in
Z_{q}(H)$ we have, $\overline{x_{2}a_{2}\wedge
y_{2}b_{2}}=\overline{x_{2}\wedge y_{2}}.$
Also,$\overline{\\{(x_{2}a_{2},x_{2}a_{2})\\}}=\overline{\\{(x_{2},x_{2})\\}\prod_{i=1}^{q-1}(x_{2}^{-1}\wedge{a_{2}}^{i})\\{(a_{2},a_{2})\\}}=\overline{\\{(x_{2},x_{2})\\}}.$
Suppose $[g_{1},g_{2}]=1$. For $h_{1},h_{2}\in H$ such that
$\alpha(\overline{g_{1}})=\overline{h_{1}}$ and
$\alpha(\overline{g_{2}})=\overline{h_{2}}$, we have
$[h_{1},h_{2}]=\beta([g_{1},g_{2}])=1$, which yields
$\phi(\overline{g_{1}\wedge g_{2}})=\overline{h_{1}\wedge h_{2}}=1.$
Thus $\phi(M_{0}^{q}(G))=1$, and we obtain a well defined map $\phi:\hskip
4.2679ptG\wedge^{q}G/M_{0}^{q}(G)\rightarrow H\wedge^{q}H/M_{0}^{q}(H)$. To
show that $\phi$ is a homomorphism, it suffices to verify that $\phi$
satisfies all the relations in the q-tensor square. We demonstrate a few below
and the others follow similarly. For each $g_{i}\in G$, let $h_{i}\in H$ with
$\alpha(\overline{g_{i}})=\overline{h_{i}}$, $i\in\\{1,2,3\\}$, and suppose
$\alpha(\overline{g_{1}g_{2}})=\overline{h}$, then we have $h=h_{1}h_{2}a$ for
some $a\in Z_{q}(H)$. Now,

$\displaystyle\phi(\overline{g_{1}g_{2}\wedge g_{3}})$
$\displaystyle=\overline{h\wedge h_{3}}$
$\displaystyle=\overline{h_{1}h_{2}a\wedge h_{3}}$
$\displaystyle=\overline{h_{1}h_{2}\wedge h_{3}}$
$\displaystyle=\overline{{}^{h_{1}}(h_{2}\wedge h_{3})(h_{1}\wedge h_{3})}$
$\displaystyle=\phi(\overline{{}^{g_{1}}(g_{2}\wedge g_{3})(g_{1}\wedge
g_{3})}).$

Let $g,g_{1}\in G$ such that $\alpha(\overline{g})=\overline{h}$,
$\alpha(\overline{g_{1}})=\overline{h_{1}}$ and
$\alpha(\overline{gg_{1}})=\overline{h^{\prime}}$. Then we have
$h^{\prime}=hh_{1}a$ for some $a\in Z_{q}(H)$.

$\displaystyle\phi(\overline{\\{g_{1}g_{2},g_{1}g_{2}\\}})$
$\displaystyle=\overline{\\{(h,h)\\}}$
$\displaystyle=\overline{\\{h_{1}h_{2}a,h_{1}h_{2}a\\}}$
$\displaystyle=\overline{\\{h_{1}h_{2},h_{1}h_{2}\\}}$
$\displaystyle=\overline{\\{(h_{1},h_{1})\\}\prod_{i=1}^{q-1}(h_{1}^{-1}\wedge(^{h_{1}^{1-q+i}}h_{2})^{i})\\{(h_{2},h_{2})\\}}$
$\displaystyle=\phi(\overline{\\{(g_{1},g_{1})\\}\prod_{i=1}^{q-1}(g_{1}^{-1}\wedge(^{g_{1}^{1-q+i}}g_{2})^{i})\\{(g_{2},g_{2})\\}}).$

The inverse map $\psi:H\wedge^{q}H/M_{0}^{q}(H)\rightarrow
G\wedge^{q}G/M_{0}^{q}(G)$ can be define similarly using $\alpha^{-1}$ and
$\beta^{-1}$. Hence $\phi$ is an isomorphism and we have the following
commutative diagram:

${1}$${B^{q}_{0}(G)}$${G\wedge^{q}G/M_{0}^{q}(G)}$${G^{q}{[G,G]}}$${1}$${1}$${B_{0}^{q}(H)}$${H\wedge^{q}H/M_{0}^{q}(H)}$${H^{q}{[H,H]}}$${1.}$$\scriptstyle{\phi}$$\scriptstyle{\beta}$

The maps $\phi$ and $\beta$ being isomorphism, yields $B_{0}^{q}(G)\cong
B_{0}^{q}(H).$ The invariance of $\widehat{B_{0}^{q}(G)}$ under
$\widehat{q}$-isoclinism can be proved similarly. ∎

######  Remark 1.

Let $G$ and $H$ be two groups, and $A\subset Z_{q}(G)$ and $B\subset
Z_{q}(H)$. If there exists a pair of isomorphisms $\alpha:G/A\rightarrow H/B$
and $\beta:G^{q}[G,G]\rightarrow H^{q}[H,H]$ with the property that whenever
$\alpha(a_{1}A)=a_{2}B$ and $\alpha(b_{1}A)=b_{2}B,$ then
$\beta([a_{1},b_{1}])=[a_{2},b_{2}]$ and $\beta(a_{1}^{q})=a_{2}^{q}$ for
$a_{1},b_{1}\in G$, for $a_{1},b_{1}\in G$. Then using the arguments in the
above theorem, we can prove that $B_{0}^{q}(G)\cong B_{0}^{q}(H)$.

######  Remark 2.

Let $G$ and $H$ be two groups, and $A\subset\widehat{Z_{q}(G)}$ and
$B\subset\widehat{Z_{q}(H)}$. If there exists a pair of isomorphisms
$\alpha:G/A\rightarrow H/B$ and $\beta:G^{q}[G,G]\rightarrow H^{q}[H,H]$ with
the property that whenever $\alpha(a_{1}A)=a_{2}B$ and
$\alpha(b_{1}A)=b_{2}B,$ then $\beta([a_{1},b_{1}])=[a_{2},b_{2}]$ and
$\beta(a_{1}^{q})=a_{2}^{q}$ for $a_{1},b_{1}\in G$, for $a_{1},b_{1}\in G$.
Then using the arguments in the above theorem, we can prove that
$\widehat{B_{0}^{q}(G)}\cong\widehat{B_{0}^{q}(H)}$.

######  Remark 3.

Note that $E_{q}^{\wedge}(G)\leqslant Z_{q}(G)$ and
$E_{q}^{\wedge}(G)\leqslant\widehat{Z_{q}(G)}$. So if $G$ and $H$ are
$q$-exterior isoclinic, then $B_{0}^{q}(G)\cong B_{0}^{q}(H)$ and
$\widehat{B_{0}^{q}(G)}\cong\widehat{B_{0}^{q}(H)}$.

##  5 Some Structural Results on q-Nonabelian Tensor Square of a Group

In this section, we generalize some structural results proved in [1] and [13].
We begin with the following lemma on the $q$-tensor square of abelian groups.

######  Lemma 5.1.

If $A$ is an abelian group, then the following statements hold:

  1. (i)

$A\otimes^{q}A$ is abelian.

  2. (ii)

If $A$ is a finitely generated group and $q$ be any non-negative integer, then
$A\otimes^{q}A$ is a finitely generated group.

###### Proof.

(i) It suffices to show that the generators of $A\otimes^{q}A$ commute with
each other. By lemma 2.2 $(iv)$, we have $[g\otimes h,g_{1}\otimes h_{1}]=\
[g,h]\otimes[g_{1},h_{1}]=1$ for all $g,g_{1},h,h_{1},\in G$. Using (5), we
obtain $[\\{(g,g)\\},\\{(h,h)\\}]=\ g^{q}\otimes h^{q}$. Now applying lemma
2.2 $(vii)$ and then (6) yields, $(g^{q}\otimes h^{q})=(g^{q}\otimes
h)^{q}=\\{([g^{q},h],[g^{q},h])\\}=1$. Furthermore, by (3) we have
$\\{(g,g)\\}(g_{1}\otimes h_{1})\\{(g,g)\\}^{-1}=\ g_{1}\otimes h_{1}$, and
hence the proof.

(ii) Let $\\{x_{i},1\leq i\leq n\\}$ be the generators of $A$ with an
ordering. We claim that $\\{x_{i}\otimes x_{j},\\{(x_{i},x_{i})\\}|1\leq
i,j\leq n\\}$ is a generating set for $A\otimes^{q}A$. Consider $a,b\in A$,
where $a=\prod\limits_{i=1}^{n}x_{i}^{m_{i}}$ and
$b=\prod\limits_{i=1}^{n}x_{i}^{l_{i}}$ for some $m_{i},l_{i}\in\mathbb{Z}$.
Since $A\otimes^{q}A$ is abelian, we have $a\otimes b=\
\prod\limits_{i=1}^{n}(x_{i}\otimes x_{i})^{m_{i}l_{i}}\prod\limits_{1\leq
i<j\leq n}(x_{i}\otimes x_{j})^{m_{i}l_{j}}(x_{j}\otimes x_{i})^{m_{j}l_{i}}$.
Applying lemma 2.2 $(ix)$, we obtain $\\{(a,a)\\}=\
\prod\limits_{i=1}^{n}\\{(x_{i},x_{i})\\}^{m_{i}}$ for $q$ odd, and when $q$
is even, using (4) and expanding as before yields the proof. ∎

Now we come to the main theorem of this section.

######  Theorem 5.2.

Let $k$ be an odd integer. If $(g\otimes g)^{k}=1$ for all $g\in G$, then
$G\otimes^{q}G\cong\nabla^{q}(G)\times G\wedge^{q}G$.

###### Proof.

Let $k=2n+1$, for some $n\in\mathbb{Z}$. Now consider the short exact sequence
$1\rightarrow\nabla^{q}(G)\xrightarrow{\alpha}G\otimes^{q}G\rightarrow
G\wedge^{q}G\rightarrow 1,$ where $\alpha$ is the inclusion map. Our aim is to
construct a left splitting map for the above exact sequence which would yield
the required direct product. Towards that, define
$\alpha^{\prime}:G\otimes^{q}G\rightarrow\nabla^{q}(G)$ on the generators as
given below.

$\displaystyle\alpha^{\prime}(g\otimes h)=$ $\displaystyle((g\otimes
h)(h\otimes g))^{-n}$ $\displaystyle\alpha^{\prime}(\\{(g,g)\\})=$
$\displaystyle(g\otimes g)^{n{{q}\choose{2}}},$

for all $g,h\in G$. Note that, for all $g\in G$, we have
$\alpha^{\prime}\circ\alpha(g\otimes g)=((g\otimes g)(g\otimes
g))^{-n}=(g\otimes g)$. Therefore it suffices to prove that $\alpha^{\prime}$
is a well-defined homomorphism. The verification of the first two relations
follows as in Theorem 4.13 of [1]. Note that from Lemma 2.2 $(i)$, we have
$\nabla^{q}(G)$ is abelian. Now since $G$ acts trivially on $\Delta^{q}(G)$,

$\displaystyle\alpha^{\prime}(\\{(g,g)\\}(g_{1}\otimes
h_{1})\\{(g,g)\\}^{-1})=$ $\displaystyle\ (g\otimes
g)^{n{{q}\choose{2}}}\big{(}(g_{1}\otimes h_{1})(h_{1}\otimes
g_{1})\big{)}^{-n}\big{(}(g\otimes g)^{n{{q}\choose{2}}}\big{)}^{-1}$
$\displaystyle=$ $\displaystyle\ \big{(}^{g^{q}}\big{(}(g_{1}\otimes
h_{1})(h_{1}\otimes g_{1})\big{)}\big{)}^{-n}$ $\displaystyle=$
$\displaystyle\ \alpha^{\prime}(^{g^{q}}g_{1}\otimes^{g^{q}}h_{1}).$

Similarly,

$\displaystyle\alpha^{\prime}\Big{(}\\{(g,g)\\}\prod_{i=1}^{q-1}(g^{-1}\otimes(^{g^{1-q+i}}g_{1})^{i})\\{(g_{1},g_{1})\\}\Big{)}$
$\displaystyle=\ (g\otimes
g)^{n{{q}\choose{2}}}\prod_{i=1}^{q-1}{{}^{g^{1-q+i}}}\big{(}(g^{-1}\otimes
g_{1}^{i})(g_{1}^{i}\otimes g^{-1})\big{)}^{-n}(g_{1}\otimes
g_{1})^{n{{q}\choose{2}}}$ $\displaystyle=\ (g\otimes
g)^{n{{q}\choose{2}}}\big{(}(g^{-1}\otimes g_{1})(g_{1}\otimes
g^{-1})\big{)}^{(-n){{q}\choose{2}}}(g_{1}\otimes g_{1})^{n{{q}\choose{2}}}$
$\displaystyle\mbox{[Lemma\ }\ref{L:1.2.3}\ \textit{(x)}]$ $\displaystyle=\
(g\otimes g)^{n{{q}\choose{2}}}\big{(}(g_{1}\otimes g)(g\otimes
g_{1})\big{)}^{n{{q}\choose{2}}}(g_{1}\otimes g_{1})^{n{{q}\choose{2}}}\ $
$\displaystyle\mbox{[Lemma}\ \ref{L:1.2.3}\ \textit{(v)}]$ $\displaystyle=\
\alpha^{\prime}(\\{(gg_{1},gg_{1})\\})\ $ $\displaystyle\mbox{[Lemma}\
\ref{L:1.2.3}\ \textit{(vi)}].$

The other relations can easily be verified, and the proof follows. ∎

The following Corollary generalizes [13, Proposition 2.12].

######  Corollary 5.3.

Let $q$ be an odd integer, then $G\otimes^{q}G\cong\nabla^{q}G\times
G\wedge^{q}G$.

###### Proof.

Using Theorem 5.2 and lemma 2.2 (ii) yields the result. ∎

######  Proposition 5.4.

If $\nabla^{q}G$ has unique $n^{th}$ roots where $n$ is even, then
$G\otimes^{q}G\cong\nabla^{q}G\times G\wedge^{q}G$.

###### Proof.

Let $n=2k$, $k\in\mathbb{N}$. Consider the short exact sequence
$1\to\nabla^{q}G\xrightarrow{\beta}G\otimes^{q}G\to G\wedge^{q}G\to 1$, where
$\beta$ is the natural inclusion map. Define
$\beta^{\prime}:G\otimes^{q}G\rightarrow\nabla^{q}G$ by

$\displaystyle\beta^{\prime}(g\otimes h)=$ $\displaystyle[(g\otimes
h)(h\otimes g)]^{\frac{k}{n}},$ $\displaystyle\beta^{\prime}(\\{(g,g)\\})=$
$\displaystyle(g\otimes g)^{{\frac{-k}{n}}{{q}\choose{2}}}.$

Proceeding as in theorem 5.2 yields $\beta^{\prime}$ is a well defined
homomorphism. Moreover, $\beta^{\prime}\beta(g\otimes g)=[(g\otimes
g)(g\otimes g)]^{\frac{k}{n}}=(g\otimes g)$, and hence the proof .

∎

## Acknowledgement

V. Z. Thomas acknowledges research support from SERB, DST, Government of India
grant MTR/2020/000483.

## References

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# The Impact of Virtual Laboratories on Active Learning & Engagement in
Cybersecurity Distance Education

Victor R. Kebande
Department of Computer Science
Blekinge Institute of Technology
Karlskrona, Sweden
<EMAIL_ADDRESS>
Use footnote for providing further information about author (webpage,
alternative address)— _not_ for acknowledging funding agencies.

###### Abstract

Virtual Laboratories ($VLabs$) have in the recent past become part and parcel
of remote teaching in practical hands-on approaches, particularly in
Cybersecurity distance courses. Their potential is meant to assist learners
with hands-on practical laboratory exercises irrespective of geographical
location. Nevertheless, adopting $VLabs$ in didactic approaches in higher
education has seen both merits and demerits. Based on this premise, this study
investigates the impact of $VLabs$ on Active Learning ($AL$) and engagement in
cybersecurity distance education. A survey with a limited number of learners
and educators who have had an experience with cybersecurity distance courses
that leveraged $VLabs$ in their practical Lab assignment, was conducted at
Blekinge Tekniska Högskola, Sweden, to assess the impact of $VLabs$ on $AL$
and engagement in Cybersecurity Distance Education. $29\%$ and $73\%$ of the
learners and educators, respectively responded to the survey administered
remotely and with good internal consistency of questionnaires based on the
Cronbalch Alpha; the results showed that learners and educators had a positive
perception of using $VLabs$ to enhance $AL$ in cybersecurity distance
education. The key concentration of the study was on $AL$ and engagement and
problem-solving abilities when $VLabs$ are used. Both the learners and
educators found the $VLabs$ to be engaging, interactive, and effective in
improving their understanding of cybersecurity concepts.

_Keywords_ Virtual $\cdot$ Laboratory $\cdot$ Distance Education

##  1 Introduction

Currently, cybersecurity has become an important and critical area of concern
that requires individuals with the necessary knowledge and skills to protect
critical infrastructures from cyber-attacks. Higher Education Institutions
play a crucial role in preparing students for careers in cybersecurity (Jung
et al., 2018). Consequently, distance education has become an increasingly
popular mode of delivering cybersecurity education due to its accessibility
and flexibility. However, distance education faces challenges in terms of
student engagement and learning, especially in the field of cybersecurity
where hands-on practical experience is crucial Khan et al. (2017); Fazza and
Mahgoub (2021).

The need to measure Active Learning ($AL$) and the use of Virtual Laboratories
($VLabs$) are two approaches that have shown the potential in enhancing
student learning in distance education. $AL$ is a student-centered approach
that emphasizes the importance of student participation and engagement in the
learning process Lumpkin et al. (2015). $VLabs$, on the other hand, provide
students with simulated hands-on experience in a controlled environment, thus
enhancing their practical skills and understanding Chiu (1999).

Despite the potential benefits of $AL$ and engagement and the use of $VLabs$,
there is a lack of an effective approach of ascertaining the impact posed by
$VLabs$ on $AL$ and engagement in cybersecurity distance education at the time
of writing this article. This is a considerable challenge as it makes it
difficult to assess effectiveness of $VLabs$ in enhancing and promoting
student learning. Consequently, assessing the suitability of $VLabs$ in
cybersecurity distance education is crucial to inform the development of
pedagogical approaches that can improve student outcomes based on their
supposed engagement.

The aim of this study is to assess the impact of $VLabs$ on $AL$ and
engagement ($AE$) in cybersecurity distance education. While the study
explores diverse aspects like $AE$ in Course Concepts ($AE$), Active Learning
and Engagement and Problem Solving ($AE$ & $PS$), difficult Course Concepts
(CC), help in Practicing Course Concepts (PCC) and Critical Thinking (CT), and
Comparison to Traditional Learning (TL). However, the focus of this study
narrows down to $AE$ and $AE$ & $PS$. The contribution of this study can be
summarised as follows:

  * •

Impact of $VLabs$ on $AL$ & engagement ($AE$) and Problem-Solving ($PS$): The
study aims to investigate the impact of Virtual Laboratories ($VLabs$) on
active learning ($AL$) and engagement ($AE$), as well as problem-solving
($PS$) in the context of cybersecurity distance education. By exploring the
potential benefits of incorporating $VLabs$ into the learning environment, the
study seeks to shed light on the effectiveness of this innovative approach in
promoting student learning outcomes.

  * •

Statistical significance: The study conducts an independent sample test to see
if there exist an statistical significance between $AE$ and $PS$ when using
$VLabs$ in cybersecurity distance education. In addition, the study examines
if there exist any relationship between $AE$ and $PS$ when using $VLabs$ in
cybersecurity distance education

  * •

Comparison with existing studies: This compares the proposed concept with
previous and existing studies as a measure of validating and reinforcing the
findings and identifying further gaps and inconsistencies if available.

This study is relevant as it addresses a significant gap in the literature on
the use of $AL$ and $VLabs$ in cybersecurity distance education. In addition
this study is important because it provide insights into how to design
effective instructional strategies and pedagogical approaches in cybersecurity
distance education and enhance student outcomes. Evaluating the impact of $AL$
when leveraging $VLabs$ proves to be an essential approach for instructors and
researchers in the field of cybersecurity education, enabling them to
determine the effectiveness of these approaches in enhancing student learning
outcomes.

The remainder of this paper is structured as follows: Section II gives the
Background, which is followed by the $VLab$ Process Model in Section III. The
methodology is given in Section IV. The Results are discussed in Section V. A
comparison with existing studies is given in Section VI while the discussions
of the proposed concepts are given in Section VII. Limitations and Future
studies are given in Section VII. Concluding Remarks are discussed in Section
VIII.

##  2 Background

###  2.1  Theoretical Framework

It is pertinent to ascertain that the targeted learners in the current
exercise are exposed to $AL$ and engagement strategies. As a result, this
study’s theoretical framework is inclined towards constructivism, which puts
the emphasis for learners to actively construct their understanding of the
subject based on exposure of some important experiences Jonassen and Rohrer-
Murphy (1999). In addition, learning in this context is viewed as a dynamic
process that involves the learner’s active engagement to cybersecurity
education Kahu (2013).

Figure 1: Theoretical Framework setting the scene of this study based on
principles of constructivism theory

By utilizing $VLabs$ as a means of assessing active learning ($AL$), this
approach aligns with the principles of constructivism theory, owing to the
fact that, it provides the learners with an opportunity to have hands-on
activities and experiences that in the long-run promotes the construction of
their knowledge. In addition, $VLabs$ are positioned to provide learners with
a collaborative and active engagement environment that is based on an
interaction with educators and other peers-which is essential in
constructivism theory because it allows the development of critical and
problem-solving skills in cybersecurity education. Figure 1 shows the role of
constructivism theory for cybersecurity which also aligns with the pedagogy
approaches as highlighted by Vygotsky Prawat (2002).

Step 1 of Figure 1 highlights the problem that learners could encounter as a
result of having or not having cybersecurity knowledge, whereas Step 2
concentrates on the effects of interactive learning and the relevance of
instructors is shown in Step 3, which ultimately leads to the learner’s
development as is shown in Step 4. While the theoretical framework shown in
Figure 1 is grounded in constructivism, its emphasis is positioned to active
learner engagement and the construction of knowledge based on meaningful
experiences in the context of; active learner engagement and knowledge
construction could include hands-on activities, problem-solving tasks, real-
world case studies, collaborative discussions, and interactive simulations.

###  2.2  Related Work

To highlight the significance of distance education in the field of
cybersecurity and the specific challenges it presents, it is observed that
distance education has become a popular mode of delivering education,
especially for cybersecurity due to its accessibility and flexibility.
However, teaching cybersecurity in a distance education setting poses several
challenges, such as ensuring students have access to hands-on learning
experiences and resources that simulate real-world scenarios. $VLabs$ offer a
solution to these challenges by allowing students to access and work on
simulated cyber attacks and defenses in a controlled environment. Studies have
shown that $AL$ is an effective teaching method in higher education, with
research indicating that it can lead to improved student engagement, learning
outcomes, and critical thinking skills Bonwell and Eison (1991), Freeman et
al. (2014). $AL$ has also been found to be particularly effective in science,
technology, engineering, and mathematics (STEM) fields, including
cybersecurity Chuang and Tsai (2018). $VLabs$ have been used in higher
education for several decades and have been shown to be effective in providing
students with hands-on learning experiences, particularly in STEM fields
Bjarnason et al. (2010). $VLabs$ are especially useful in distance education
as they allow students to access and work on simulated scenarios, providing a
controlled environment for learning that would not be possible in the real-
world ( Huang et al. (2019).Studies have also shown that $VLabs$ can be used
effectively in cybersecurity education, providing students with opportunities
to work on simulated cyber attacks and defenses in a controlled environment
Kapetanakis et al. (2016). In the context of cybersecurity distance education,
$VLabs$ can provide students with hands-on experiences that would not be
possible in a traditional classroom setting Tsai et al. (2020). However, there
is limited research on the impact of AL and engagement activities centered in
$VLabs$ with particular interest to student learning in cybersecurity distance
education Jung et al. (2018). Therefore, there is a need for approaches that
can assess the effectiveness of this approach to teaching and learning.

##  3 VLabs process Model

Virtual Laboratories ($VLabs$) sequences provides an integral process that
allows learners to be able to have a hands-on approach, with an interactive
environment that is supported by remote execution of practical assignment over
emulations of computers in cybersecurity distance education. $VLabs$ can
support multiple task, share data, and real-time collaborative execution of
hands-on practical assignments. Figure 2 shows $Vlab$ processes that are
$labelled$ $1$ to $5$. The Educator Module (EM) $labeled$ $1$ allows the
instructor to create the $VLab$ exercises. It serves as as the genesis of the
preparation of course materials and instructions to learners which is done in
the step $labeled$ $2$ with the aid of provisioning a $Virtual$ $Machine$ (VM)
over a Virtual Space Interaction (VSI). A $VM$ is a software emulation of a
physical computer that allows multiple Operating Systems (MOS) to run
simultaneously on a single computer. Once the $VLab$ exercise is ready,
learners are able to receive instructions and guidelines from the educators
regarding the objectives, tasks, and resources available in the $VLabs$.

Figure 2: Sequences in the $VLabs$ processes Momen and Kebande (2023).

This interaction enables the learners to be able to work with a logical
abstraction of a computer and allows them to inquire, seek clarification and
execute tasks and receive real-time feedback as they engage with the $VLab$. A
Cyber-Constructive Nexus (CCN) in the step $labeled$ $3$ provides a link
between the VSI and the virtualized pace ($labeled$ $4$). Through the
learner’s module in the step $labeled$ $5$, learners interact with the $VLab$
through a VM. They access the VLab platform, where they can perform hands-on
activities, experiments, and simulations related to cybersecurity concepts and
practices in readiness for assessment. This hands-on experience allows
learners to apply theoretical knowledge, develop practical skills, and gain a
deeper understanding of the subject matter. Throughout the learner’s
engagement with the $VLabs$, there is a continuous exchange of instructions
and communication between the educator and the learner in the step $labeled$
$6$ and $6$, however, not in real-time in some cases.

Figure 3: Proportion of learners that feel Active learning and engagement has
increased interest in the course

##  4 Methodology

In order to assess the impact of $VLabs$ in cybersecurity distance education,
a three-fold methodology has been leveraged encompasing, sampling and face
validation, assessment and testing. The three-fold approach has been used to
assess the learners and educators outcomes based on purposive sampling where a
list of learners who had experienced the use of $VLabs$ was considered, which
included face validity approaches that allowed cybersecurity experts to give
expert reviews based on the prevailing research question. This study is cross-
sectional, owing to its collection of data at a single point in time from
different individuals and groups in order to examine relationships or
differences. Also, the study leverages a quasi-experimental approach with pre-
test that have been used to test the significance. Other subprocesses that
have been used in the study are explained in the subsequent sections.

Table 1: Learners’ and Educators’ Questionaire (Cronbach Alpha) Target | Cronbach Alpha
---|---
Learners Questionnaire | 0.791
Educator’s Questionnaire | 0.847

###  4.1 Context and Participants Selection

The participants consisting of learners, were recruited from the Department of
Computer Science at Blekinge Institute of Technology, Sweden. A random and
voluntary survey with questions shown in Table 2 was passed to learners who
currently or previously had leveraged $VLabs$ in Cybersecurity Distance
Education in the past 3 years. The participants constituting educators who
have had experiences with $VLabs$ were non-randomly recruited using purposive
sampling recruited from BTH, Sweden, Malmö University, Sweden, Curtin
University, Australia, Halmstad University, Sweden, and Zayed University, UAE.
A survey with question shown in Table 3 was passed to the educators in order
to assess theor perceptions on use of $VLabs$. In addition, there was no
significance of identifying the participant’s age, gender and ethnicity among
learners and educators in this study.

Figure 4: Proportion of Educators that feel Active learning and engagement has
increased interest in the course

###  4.2 Test Surveys

The survey aimed to assess $VLabs$’ impact on $AL$ and engagement in
Cybersecurity Distance Education. A relatively, similar set of questions was
administered to both the learners and educators. Six (6 ) items were assessed
using a 5-point Likert scale ranging from "Strongly Disagree-1" to "Strongly
Agree-5".

Table 2: Administered Learners Questionnaire Survey question | Scale
---|---
Do you feel that the use of $VLabs$ have helped you understand difficult concepts in your course? | 1-5
Do you feel that the use of $VLabs$ have have improved your AL & engagement and interest in the course material? | 1-5
Is the use of $VLabs$ in distance courses most appropriate when compared to traditional (in-person) laboratories? | 1-5
Do you feel that $VLabs$ have allowed you to apply the course concepts in practical setting? | 1-5
Do you feel that using $VLabs$ have improved your thinking skills? | 1-5
Do you feel that taking practicals virtually allows you to actively learn & engage in problem-solving, discussions, and |
hands-on activities well as compared to traditional approaches? | 1-5
Table 3: Administered Educators Questionnaire Survey question | Scale
---|---
Have you used $VLabs$ in giving cybersecurity/online/distance courses? | Y/N
Do you feel that giving $VLabs$ has helped distance students to understand the difficult concepts in the course? | 1-5
Do you feel that $VLabs$ have improved distance students AL & engagement and interest in the course materials? | 1-5
Is the use of $VLabs$ in distance courses most appropriate when compared to traditional (In-person) laboratories? | 1-5
Do you feel that giving $VLabs$ to distance students have allowed students |
to apply the course concepts in practical setting? | 1-5
Do you feel that giving $VLabs$ have improved student thinking skills? | 1-5
Do you feel that distance students taking practicals virtually do actively learn engage in problem-solving, |
discussions, and hands-on activities well as compared to traditional approaches | 1-5 .

There were four subscales namely: Course concepts, active engagement,
comparison to traditional approaches, and critical thinking skills. Two (2)
items were related to the general learning course concepts, while two (2)
items had a relationship to $AL$ and engagement and one (1) item was inclined
to the preference of $VLabs$ to traditional approaches. In addition to this,
three (3) items focused on applying the skills based on the constructivism
theory Jonassen and Rohrer-Murphy (1999), as was mentioned in Section II of
this paper. Among the 70 learners who were randomly selected to participate in
the survey, 20 consented to participate in this activity, and observations
were made from the 20 responses, representing approximately a response rate of
29%. Consequently, 11 out of 15 educators responded with a response rate of
approximately 73%. From the 6 items that were assessed for the learners and
educators, a Cronbach’s Alpha of 0.791 and 0.847 were realized respectively as
is shown in Table 1, which indicates a good level of internal consistency and
relatively high reliability given that Cronbach’s alpha coefficient of 0.7 or
higher is considered acceptable. The maximum possible score that could be
achieved in the assessment was 30, a score above the median is regarded as a
good engagement with $VLabs$ among learners which implies learners appreciate
the concepts in $VLabs$.

Table 4: Descriptive statistics for the Educators  Dimension | AE |  $AE$ &$PS$ | CC | TL | PCC | CT
---|---|---|---|---|---|---
N | 11 | 11 | 11 | 11 | 11 | 11
Mean | 3.91 | 3.27 | 3.91 | 3.45 | 4.18 | 4.27
Median | 4.00 | 3.00 | 4.00 | 4.00 | 5.00 | 4.00
SD | 1.044 | 1.421 | 1.136 | 1.128 | 1.250 | .786
Variance | 1.091 | 2.018 | 1.291 | 1.273 | 1.564 | .618
Skewness | -1.074 | -.074 | -.789 | -.118 | -1.912 | -.574
Kurtosis | .581 | -1.384 | -.546 | -1.306 | 3.867 | -.967
Table 5: Descriptive statistics for the Learners  Dimension | $AE$ |  $AE$ &$PS$ | CC | TL | PCC | CT
---|---|---|---|---|---|---
N | 20 | 20 | 20 | 20 | 20 | 20
Mean | 3.30 | 3.05 | 3.25 | 3.40 | 3.25 | 4.27
Median | 4.00 | 3.00 | 4.00 | 3.50 | 5.00 | 4.00
SD | 1.525 | 1.538 | 1.517 | 1.603 | 1.482 | 1.603
Variance | 2.326 | 2.366 | 2.303 | 2.568 | 2.197 | 2.568
Skewness | -.566 | -.382 | -.370 | -.396 | -.263 | -.713
Kurtosis | -1.094 | -1.512 | -1.36 | -1.306 | -1.476 | -.1.082

###  4.3 Data Collection

The survey encompassing questionnaires shown in Tables 2 and 3 respectively
was distributed to learners and educators via university student and staff
email between February and March 2023. In order to recruit participants, the
survey was created using Google Forms and it was sent using as a link to
participants twice in a span of one month. Informed consent that provided
information regarding the survey stated that the study was voluntary and
anonymous and the responses were kept confidential. It also stated that the
participants were free to withdraw at will. Furthermore, the consent also
highlighted that the outcome/result from the survey was explicitly to inform
the impact of $VLabs$ on $AL$ and engagement in cybersecurity distance
education. Furthermore, it is worth mentioning, even though the study was
anonymous, only the study investigator had access to the participant’s data,
and this data was securely stored.

###  4.4 Data Analysis

IBM Statistical Package for Social Sciences (SPSS) V28 was used to analyze
data after data had been cleaned. A probability of <0.05 was considered to be
significant for the test conducted (alpha=0.05). This involved descriptive
statistics that allowed calculation of the central tendencies; Mean, median,
Standard deviation,variance, Skewness, Kurtosis and analysis of descriptive
variables for both the learners and educators in Cybersecurity distance
education as is shown in Tables 4 and 5 respectively. The considered aspects
in this context included $AL$ and Engagement in Course Concepts ($AE$), Active
Learning and Engagement and Problem Solving ($AE$ & $PS$), difficult Course
Concepts (CC), help in Practicing Course Concepts (PCC) and Critical Thinking
(CT), and Comparison to Traditional Learning (TL), however, the focus of this
study was inclined towards $AE$ and $AE$ & $PS$. In addition, inferential
statistics were used to test hypotheses and determine statistical
significance. The results of the data analysis will provide insights into the
perceptions of $AL$ and engagement among cybersecurity distance learners and
inform future research and practice in this field.

##  5 Results

###  5.1 Demographics

Twenty learners (20) learners and 11 educators participated in this study
which amounted 29% and 73% respectively. Learners were selected randomly, from
the Network and System Security Course, which is part of Msc Telecommunication
Programme at Blekinge Institute of Technology, BTH, Sweden. Of the
participants, 63% of the educators responded to have used $VLabs$ in giving
cybersecurity distance courses, 9% of the educators were not sure and 27% had
not used $VLabs$. In the $AL$ and engagement on the course, 36.4% of the
educators and 25% of the learners had a strong perception about the effect of
$VLabs$ as is shown in Figure 3 and Figure 4 respectively. The significant
aspects that were observed aspects like $AL$ and Engagement in Course Concepts
($AE$), Active Learning and Engagement and Problem Solving ($AE$ & $PS$),
difficult Course Concepts (CC), help in Practicing Course Concepts (PCC) and
Critical Thinking (CT), and Comparison to Traditional Learning (TL), however,
the focus of this study narrows down to $AE$ and $AE$ & $PS$ which are
discussed in Sections 5.2 and 5.3 respectively. Although all the items ($AE$,&
$PS$,CC,PCC,CT and TL are studied and included in the study as part of the
responses from the learners and educators, it is worth mentioning that, the
motive of exploring this concepts is that, they have been used to set the
stage for the specific aspects that forms the major part of the study of this
paper ($AE$ & $PS$). It should also be noted that the inclusion of these
concepts do not influence the outcome of the study. This narrowed focus allows
for a more in-depth analysis of the effects of VLabs on these specific
dimensions of learning and engagement in the context of cybersecurity
education.

###  5.2  AL and Engagement on the course

In examining the learner’s and educator’s impact on engagement and interest of
the course when $VLabs$ are used showed the following was observed; 25% of the
learners strongly agreed that the use of $VLabs$ have had a positive impact on
the course, which included improved engagement and interest in the course
material. Also, 25% strongly disagreed, 30% somewhat agreed to have seen the
benefits and 20% were neutral. On the same note, 36.4% of the educators agreed
that using $VLabs$ has had a huge impact on student engagement and interest to
the course with 36.4% somewhat agreeing,18.2% somewhat disagreed, with 9.1%
staying neutral. The responses from learners and educators on the use of
$VLabs$ and Engagement on the course reported (M=3.30, SD=1.525 and 3.91 and
1.044 respectively)

###  5.3  AL and Problem-Solving

Descriptive statistics for the impact of $VLabs$ on AL and engagement was
meant to assess whether $VLabs$ actively helps the students in their problem-
solving skills. 30% of the learners said that the use of $VLabs$ had not
allowed them to actively engage in problem-solving, having key study
discussions and helping them with hands-on activities. 40% somewhat agreed
that the use of $VLabs$ had helped. Only 15% of the learners attributed
$VLabs$ to have strongly helped them with their active engagement in problem-
solving abilities. 10%, however, stayed neutral with 5% somewhat disagreeing.

Likewise, 27.3% of the educators strongly agreed that $VLabs$ had improved
active engagement and problem-solving abilities for learners with 18.2% of the
educators somewhat agreeing to this. However, 27.3% of the educators somewhat
disagreed with this and 9.1% strongly disagreed with this. 18.2% of the
educators chose to be neutral. The responses from learners and educators on
the use of $VLabs$ to assess the ability of problem solving in the course
reported (M=3.05, SD=1.538 and 3.27 and 1.421, respectively)

###  5.4 Independent T-test Analysis

In order to check the statistical significance between $AL$, engagement and
problem-solving among the learners and educators responses, an independent
T-test was conducted in order to analyze the data from the two groups as is
shown in Table 6. Independent t-test was chosen as the appropriate statistical
test that could compare the means of the variables in those groups. The
hypothesis that guided this analysis was as follows:

Table 6: Independent Sample Test Dimension | Var | F | Sig | t | df | 1-sp | 2-sp | MD | S.E D | Lw | Up
---|---|---|---|---|---|---|---|---|---|---|---
Active Learning in CM | EV Assm | 2.917 | 0.131 | -7.514 | 7 | <.001 | <.001 | <-3.667 | <0.488 | <-4.820 | <-2.513
- | EV Not Assm | - | - | -11.000 | 5.000 | <.001 | <.001 | <-3.667 | <.333 | <-4.524 | <-2.810

Ho: There is no difference in the means of active learning and engagement, and
problem solving between the Learners and Educators

The purpose of this hypothesis was to determine whether there exist any
significant or statistical differences in the levels of $AL$ and engagement
and problem-solving between the two groups. The independent T-test analysis
mainly covered two independent variables from the learner’s group as is shown
in 6. The dimension showing Active Learning in CM, shows the $F-Value$ of
2.917 and a corresponding p-value of 0.131. The t-value shows -7.514 and a
degree of freedom (dof) of 7.

The one-sided p-value (1-sp) and two-sided $p-value$ (2-sp) are both less than
0.001. In addition, the Mean Difference (MD) was calculated as -3.667 with s
Standard Error (S.E) of 0.488. The lower (Lw) and Upper (Up) confidence
intervals are given as -4.524 and -2.810 respectively. The independent t-test
gives $F-test$ based on Levene’s test of equality of variances was used to
check if the variance is equal. Since the study do not have an alternative
hypothesis, it was observed that our significance for p-value for $F-test$ is
less than $alpha$= $(0.05)$, indicating that the variance are not assumed to
be equal since the p-value is <0.05, the groups that had a response on $AL$
and problem-solving are not assumed to be the same.

The results from the independent-t-test provide differences in terms of $AL$,
engagement, and problem-solving among learners and educators. The significant
negative $t$ value shows that there exist statistical significant differences
of the variables from the two groups. Therefore having a p-value $0.01<0.05$
than the initially chosen significance level of 0.05 show that the null
hypothesis, Ho falls on the regions of rejection.

From the findings, conclusions can be made that there exist significant
differences in the means of $AL$ and engagement, and problem-solving among
learners and educators. This shows that the learners and educators have
distinct levels of active learning, engagement, and problem-solving as far as
cybersecurity distance courses are concerned.

###  5.5 Correlation Analysis

A correlation analysis meant to examine the relationship between $AL$ and
engagement, and problem-solving among learners and educators was conducted as
is shown in Table 7 and 8 respectively.

Table 7 shows the correlation coefficients between $AL$ and engagement ($AE$)
and problem-solving ($AE$ & $PS$) for the learners. The Pearson correlation
coefficient $(r)$ was used to determine the strength and direction of the
relationship, while the p-value was used to assess the statistical
significance of the correlation. The Pearson correlation coefficient between
$AL$ and engagement is 0.891. This indicates that there is a strong positive
correlation among these variables in learners with $(r=0.891,p<0.05)$ as is
shown in Table 7. The correlation is statistically significant $(p<0.001)$,
which indicates that an increase in $AL$ so does the level of engagement among
the learners. Also, the correlation coefficient for $AE$ & $PS$ is 0.891 which
also indicates a strong positive correlation. Consequently, it is also
statistically significant at $(p<0.001)$ showing that an increase of $AL$
triggers the increase the level of engagement and problem-solving skills among
learners.

Table 7: Learner’s correlations’ $VLabs$ engagement and Active Engagement and problem solving Correlations outcomes | $AE$ |  $AE$ &$PS$
---|---|---
$AE$ | - | -
Pearson r | 1 | 0.891
Sig. (two-tailed) |  | <.001
N | 20 | 20
$AE$ &$PS$ | - | -
Pearson r | 0.891 | 1
Sig. (two-tailed) | <.001 | -
N | 20 | 20
Correlation is significant at 0.001 level (two-tailed)
Table 8: Educator’s correlations’ AL & engagement and Active Engagement and problem solving Correlations outcomes | $AE$ |  $AE$ &$PS$
---|---|---
$AE$ | - | -
Pearson r | 1 | .423
Sig. (two-tailed) | - | .195
N | 11 | 11
$AE$ &$PS$ |  |
Pearson r - | .423 | 1
Sig. (two-tailed) |  | <.195
N | 11 | 11

On the other hand, Table 8 shows the correlation coefficients between $AE$ and
$AE$ and ($AE$ & $PS$ for educators. As is shown in Table 8, the Pearson
correlation, r between the variables is $(r=0.423,p<0.05)$, which indicates a
moderate positive correlation. This correlation is not statistically
significant $(p=0.195)$. This suggested that there may not exist a strong
relationship between the $AL$ and engagement among the educators. Also, the
correlation coefficient for $AE$ & $PS$ among the educators is 0.423
indicating a moderate correlation, which is not statistically significant
$(p=0.195)$, which suggests that there may not exist a significant
relationship between $AL$ and engagement, and problem-solving skills among the
educators.

The findings from this correlation analysis provide insights on the
relationship between $AL$, and engagement and problem-solving among learners
and educators. It has been observed that among the learners, there exist a
strong positive correlation among $AL$, engagement and problem solving. This
shows that as the learners actively engage in the learning process, their
engagement levels and problems solving skills do increase too. However, that
is not the case among the educator’s perception; the correlation between $AL$,
and engagement and problem-solving is not statistically significant-which
shows that there is no strong relationship how they feel about these
variables.

##  6 Comparison with existing studies

In this section a comparison that assesses the impact of the proposed approach
with existing studies is conducted in the context of cybersecurity distance
education. The motivation of exploring the comparative approach is to seek to
examine and evaluate the similarities and differences that exist in the
proposed/observations against the literature. Using surveys administered
encompassing questionnaires shown in Tables 2 and 3 respectively to a limited
number of learners and educators with experience in cybersecurity distance
courses.

A study of $VLabs$ environment as a method of educating cybersecurity
researchers that has been highlighted by Nance et al. (2009) shows a
significant impact. Through this study, unique implementations are assessed
from local to remote access. The findings show the need for having a
significant amount of resources to create a virtual study environment. While
this study remains relevant, it is presented from a generic perspective, given
that it is not centered on student learning and engagement. Also, a
personalized learning perspective in a Virtual Hands-on lab platform for
Computer Science Education is able to identify individual student learning
styles during lab sessions. This study went an extra mile to automatically
assess, predict and monitor student learning and performance changes over
time. While this approach was able to show a positive impact and efficiency on
the student’s, there was no measure on the degree of engagement Deng et al.
(2018). Another study that analyzed the students that had to learn in a
container-based $VLabs$ for cybersecurity assessed the factors that influenced
student learning, where indicators that were relevant to the student behavior
on courses were assessed like estimated Effort (EE), Perceived Usefulness
(PU), social influence, usefulness and intension of use. While this study was
also relevant, it hardly had a focus on Active Engagement Robles-Gómez et al.
(2019). An assessment of student learning in the virtual environment in
computer networks course reveals that in order for students to learn in a
virtual laboratory, prior experience counts and the study distinguished the
learning events in lectures and laboratories. The study is insightful as far
as the effectiveness is concerned, however, it was viewed from the students’
perspective Wolf (2009). Other relevant studies on the use of virtual labs
include using $VLabs$ network testbed technology, which showed a positive
effect on student knowledge, especially student thinking, when aligned to
Bloom’s taxonomy Luse and Rursch (2021). Also, the learners, support services
can affect student engagement when online Learning environments are used as
highlighted by He et al. (2019). The findings suggest that when instructors
use online learning Services (OLS) as indicators, it assists in understanding
the learning processes among students.

##  7 Discussions

The findings from this study have provided insights on the positive impact of
$VLabs$ on $AL$ and engagement in cybersecurity distance education. The survey
results have revealed that both learners and educators had a favorable
perception of the use of $VLabs$ in enhancing their understanding of
cybersecurity concepts. The following sections discuss the key findings and
their implications in more detail.

Based on the sample from the learners and the educators, this study has been
able to examine the impact of $VLabs$ on $AL$ and engagement in Cybersecurity
distance education. It was observed that a proportion of learners strongly
agreed that using $VLabs$ in cybersecurity distance courses not only actively
enabled them to engage in the course but also it had an effect in their
problem-solving skills. A good proportion of learners agreed that $VLabs$ is a
worthy approach as far as distance education is concerned. In addition, they
found that to some extent, the $VLabs$ helped them to improve their practical
skills in cybersecurity. However, a proportion of the learners were not
certain but chose to stay neutral on this with some disagreeing.

A percentage of respondents who agreed that $VLabs$ had a positive impact on
their engagement and interest in the cybersecurity distance course is
noteworthy. This suggests that among learners the $VLabs$ provide an effective
means to actively engage with the course material, problem-solving strategies
resulting in continuous active engagement. The hands-on nature of $VLabs$
allow learners to apply theoretical knowledge in practical scenarios,
fostering a deeper understanding of cybersecurity principles. It has been
observed also that the positive impact on engagement is important in distance
education settings where learners may face challenges of continuous
interaction with the course content.

It has also been observed that $VLabs$ facilitates active engagement in
problem-solving among learners. This is because, a significant proportion of
respondents acknowledged that $VLabs$ had helped them actively engage in
problem-solving activities, discussions, and hands-on exercises. This outcome
consolidates the notion that $VLabs$ provide a a suitable platform and
environment for learners critically think and be equiped with problem-solving
capabilities.

It is worth to note that although a small percentage of respondents had a
neutral perception of the impact of $VLabs$, it was still seen that a good
proportion considered the use of $VLabs$ to be essential. A number of factors
that could lead to this could be attributed to lack of prior experiences or
customed learning preferences or technical challenges. It is the authors
opinion that providing guidance and resources for learners to navigate and
utilize $VLabs$ effectively and instructors role could overcome the prevailing
concerns.

Literature has also shown the need for active engagement for learners when
utilizing remote resources. One of the key aspects that stimulated student
engagement is the course materials, which act as a determinant on many
learning outcomes. Since both the learners and educators had both interacted
with $VLabs$ environment, it is the author’s opinion that it could be
important for the educators to initiate key strategies that could ensure that
there is continuous engagement among the learners.

Consequently, correlation results show that the learner’s perception on
$VLabs$ engagement was significantly high coupled with problem-solving
abilities when compared to the perception of the educators. The most ideal or
probable reason for this can be attributed to the respondents having previous
experience with $VLabs$ environment or respondents that are mainly from
Computer Science discipline.

It is also worth noting that the suggestions in this study are also aligned to
the ICAP framework Chi and Wylie (2014), as mentioned in Section II of this
paper, which shows an emphasis on how $AL$ and student engagement promotes
student learning. These claims can be supported by the scores shown from the
outcome of this study.

##  8 Limitations and Future Studies

While important insights have been pointed out in this study , it is worth
acknowledging that there exist limitations and and it is also worth
considering the avenues for future studies. The sections that follow discuss
the limitations that were encountered during the study and a suggestion of
future direction is given.

One limitation of this study is the use of relatively a small sample size of
participants especially for the educators. Even though efforts were made to
have a diverse range of learners and educators who had experienced
cybersecurity distance courses leveraging $VLabs$, the sample size do not
fully represent the entire population. For future studies it could be
beneficial using larger sample sizes and wider participant recruitment in
order to enhance the generalizability of our findings.

There is also a limitation relying on the reported data from survey responses.
While the author agrees that surveys provide a convenient means of gathering
data, they could have potential biases. Future studies could incorporate
additional and relevant research methods, in order to gain a more harmonized
and comprehensive understanding of the impact of $VLabs$ on $AL$ and
engagement in cybersecurity distance education.

While it is important to note that the outcomes that that were expressed by
participants are valuable, future research is projected to incorporate
examination assessments in order to objectively measure whether the
effectiveness of $VLabs$ in improving learning outcomes in cybersecurity
distance education can have an influende in pedagogical practices.

##  9  Conclusions

The aim of this study was to investigate the impact of $VLabs$ on $AL$ and
engagement in cybersecurity distance education. From a survey of learners and
educators that had experience in cybersecurity distance education that
utilized $VLabs$, a survey was conducted and important insights have been
reported. The findings of this study have shown that $VLabs$ are viewed
positively by both learners and educators, as they enhance active engagement
and problem solving strategies. Furthermore, the outcome has shown that
$VLabs$ have the potential to enhance $AL$ in cybersecurity distance
education.

## Declaration

Data availability used in this research is available on request and the author
has no competing interests.

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