#  Finite split metacyclic groups and their 2-nilpotent multipliers

S. Aofi Al-Akbi and S. Hadi Jafari
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad,
Iran

###### Abstract

There has been a great importance in understanding the nilpotent multipliers
of finite groups in recent past. Let a group $G$ be presented as the quotient
of a free group $F$ by a normal subgroup $R$. Given a positive integer $c$,
the $c$-nilpotent multiplier of the group $G$ is the abelian group
$\mathcal{M}^{(c)}(G)=(R\cap\gamma_{c+1}(F))/\gamma_{c+1}(R,F)$, where
$\gamma_{1}(R,F)=R$, $\gamma_{c+1}(R,F)=[\gamma_{c}(R,F),F]$, and
$\gamma_{c+1}(F)=\gamma_{c+1}(F,F)$. In particular, $\mathcal{M}^{(1)}(G)$ is
the Schur multiplier of $G$. The crucial aspect of the research in to the
$c$-nilpotent multipliers of groups includes either establishing their
structures, or estimating their sizes and exponents. One reason for studying
the $c$-nilpotent multiplier is its relevance to the isologism theory of P.
Hall. The study of Schur multiplier of finite metacyclic groups goes back to
the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent
multiplier of finite split metacyclic groups with the help of their nonabelian
tensor squares. In particular, we give a complete description of the triple
tensor product, the triple exterior product, and the 2-nilpotent multiplier of
such groups.
Keywords Metacyclic group . Group action . Nonabelian tensor product .
Nilpotent multiplier
Mathematics Subject Classification 20F05 . 20C25 . 20J99

##  1 Definitions And Motivation

A group $G$ is said to be metacyclic if it possesses a cyclic normal subgroup
$N$ such that $G/N$ is also cyclic. It is clear that a metacyclic group $G$
can be written $G=HN$ with $H\leqslant G,$ $N\unlhd G$ and both $H$ and $N$
cyclic. If $H\cap N=1,$ then $G$ is called the split metacyclic group. It was
already known to Hölder that finite metacyclic groups can be presented on two
generators and three defining relations. When $G=HN$ is split, by taking
generators $x$ of $H$ and $y$ of $N,$ it is well-known that $G$ has a
presentation of the form

$\displaystyle\langle x,y~{}|~{}x^{m}=y^{n}=1,~{}xyx^{-1}=y^{l}\rangle,$ (1)

where the integers $m,n,$ and $l$ satisfying $l^{m}\equiv 1$ (mod $n$). In the
rest of the paper we will fix $m,n,$ and $l$ as the integers of the above
presentation. The purpose of this paper is to show that the $2$-nilpotent
multiplier $\mathcal{M}^{(2)}(G)$ is the direct product of two cyclic groups
of orders $(n,l-1,1+l+...+l^{m-1})$ and $1\over
n$$(n,1+l+...+l^{m-1})(n,(l-1)^{2}),$ where by $(r,s)$ we mean the greatest
common divisor of the integers $r$ and $s$.

Given a positive integer $c$, recall that the $c$-nilpotent multiplier of a
group $G$ which is presented as the quotient of a free group $F$ by a normal
subgroup $R$, is the abelian group
$\mathcal{M}^{(c)}(G)=(R\cap\gamma_{c+1}(F))/\gamma_{c+1}(R,F),$ where
$\gamma_{1}(R,F)=R$, $\gamma_{c+1}(R,F)=[\gamma_{c}(R,F),F]$ and
$\gamma_{c+1}(F)=\gamma_{c+1}(F,F).$ It was shown by R. Baer [1] that these
invariants are, up to group isomorphism, independent of the choice of free
presentation of $G$. The group $\mathcal{M}(G)=\mathcal{M}^{(1)}(G)$ is more
known as the Schur multiplier of $G$.

One motivation to study the $c$-nilpotent multipliers is its relevance to the
isologism theory of P. Hall [8, 9] in which groups can be classified into
isologism classes (see [5, p. 406] for more details). Another important reason
is that, determining the structures of $c$-nilpotent multipliers is essential
in studying generalized capability and covering groups, and would be generally
useful in developing such a framework. Furthermore, although the above formula
of $\mathcal{M}^{(c)}(G)$ looks simple enough; it is mostly a hard but
interesting problem to compute it. In particular, according to the method
proposed in [5], which is based on the notion of triple exterior product, we
hope our different approach of studying 2-nilpotent multipliers can be
extended to more general cases of c-nilpotent multipliers and provides some
new insights.

Let’s first recall the definition of the nonabelian tensor product of two
groups $G$ and $H$, by supposing that they act on each other satisfying
certain compatibility conditions [3, pp. 178-179], and act on themselves by
conjugation. The nonabelian tensor product $G\otimes H$, was introduced by
Brown and Loday in [4], is defined as a group generated by the symbols
$g\otimes h$, where $g\in G$ and $h\in H$, with defining relations

$gg^{\prime}\otimes h=({{}^{g}}g^{\prime}\otimes{{}^{g}}h)(g\otimes h)\hskip
14.22636pt\text{and}\hskip 14.22636ptg\otimes hh^{\prime}=(g\otimes
h)(^{h}g\otimes{{}^{h^{\prime}}}h),$

for all $g,g^{\prime}\in G$ and $h,h^{\prime}\in H$, where
${}^{g}g^{\prime}=gg^{\prime}g^{-1}$. It was shown in [6] with homological
methods, involving the original approach in [3, 4], that $G\otimes H$ is a
finite group, when $G$ and $H$ are finite groups.

Since the compatible actions are satisfied when a group acts on itself by
conjugation, it leads to define the nonabelian tensor square $G\otimes G$. The
nonabelian exterior square $G\wedge G$ is the quotient of $G\otimes G$ modulo
the central subgroup $\nabla(G)$ generated by the elements $g\otimes g$ for
all $g\in G$. The image of the generator $g\otimes g^{\prime}$ in $G\wedge G$
is written $g\wedge g^{\prime}$. The commutator map induces a homomorphism
$[~{},~{}]:G\wedge G\longrightarrow G$ mapping $g\wedge g^{\prime}$ to
$[g,g^{\prime}]=gg^{\prime}g^{-1}g^{\prime-1}$, in which its kernel is
isomorphic to $\mathcal{M}(G)$ [12].

The first attempt at determining the tensor square of a finite split
metacyclic group $(1)$ was made by Brown, Johnson, and Robertson [3] where
they achieved the favorable special case when $n$ is odd.

######  Proposition 1.1.

([3, Proposition 15]) Let $G$ be a metacyclic group with presentation $(1)$
and $n$ is odd. Then $G\otimes G$ is the direct product of four cyclic groups
with generators $x\otimes x,\ y\otimes y,\ (x\otimes y)(y\otimes x),\ x\otimes
y,$ $of\ orders$ $m,\ (n,\ l-1),\ (n,\ l-1,\ 1+l+...+l^{m-1}),\ and\ (n,\
1+l+...+l^{m-1}),$ $respectively.$ Furthermore, $\nabla(G)$=$\langle x\otimes
x,\ y\otimes y,\ (x\otimes y)(y\otimes x)\rangle$ and $\mathcal{M}(G)$ is
cyclic of order $\frac{1}{n}(n,l-1)(n,1+l+...+l^{m-1})$.

Next Johnson [11] completed such a computations be evaluating $G\otimes G$ for
even $n$.

######  Proposition 1.2.

([11, Proposition 16]) Let $G$ be a metacyclic group with presentation $(1)$
and $n$ is even. Then $G\otimes G$ is the abelian group with generators
$X=x\otimes x,$ $Y=y\otimes y,$ $Z=(x\otimes y)(y\otimes x),$ $T=x\otimes y,$
and $A=(y\otimes y)^{l-1},$ and relations

$\displaystyle X^{m}$
$\displaystyle=Y^{n}=Z^{l-1}=Z^{n}=Z^{1+l+...+l^{m-1}}=T^{n}=A^{2}=A^{m}=1,Y^{l-1}=A,\
T^{1+l+...+l^{m-1}}=A^{(l-1)(m-1)m/4}.$

Furthermore, $\mathcal{M}(G)$ is cyclic of order
$\frac{1}{n}(n,l-1)(n,1+l+...+l^{m-1})$.

With the above assumptions, one could easily observe that $\nabla(G)\cong
C_{m}\times C_{(n,2(l-1))}\times C_{(n,l-1,1+l+...+l^{m-1})}.$

Here we need to give a brief recall of the notion of triple tensor (exterior)
product of groups. The notation and terminology are the same as those in [10].
Since conjugation yields compatible actions, there is a diagonal action of $G$
on $G\otimes G$ such that ${}^{g}(g_{1}\otimes
g_{2})=~{}{{}^{g}}g_{1}\otimes{{}^{g}}g_{2}$. Also the tensor square $G\otimes
G$ acts on $G$ by conjugation in $G$ via the homomorphism $[~{},~{}]:G\otimes
G\longrightarrow[G,G]$ induced by the commutator map. Evidently these actions
are compatible and we can thus construct the triple tensor product
$\otimes^{3}G=(G\otimes G)\otimes G$. Also by [4, Definition 2.11] the triple
exterior product $\wedge^{3}G=(G\wedge G)\wedge G$ is obtained from the tensor
product $(G\wedge G)\otimes G$ by imposing the additional relations $(g\wedge
g^{\prime})\otimes[g,g^{\prime}]=1$ for all $g,g^{\prime}\in G$ (see [10] for
more details). Recently, the second author in joint work with some others [7,
10] established the explicit structures of the triple tensor (exterior)
products of certain groups.

Like the above results, we’ll see below that although the structures of
$\wedge^{3}G$ and $\mathcal{M}^{(2)}(G)$ remain invariant for any integer $n$,
but the evaluation of $\otimes^{3}G$ for even $n$ is somewhat different from
the odd $n$. Because for even case the computations depend on the relation
$T^{1+l+...+l^{m-1}}=A^{(l-1)(m-1)m/4}$ which appeared above. The main result
of this paper is the following theorem.

Main Theorem. Let $G$ be a finite split metacyclic group with presentation
$(1)$.

$(i)$ If $n$ is odd, then

$\otimes^{3}G\cong C_{m}\times C_{(n,l-1)}\times C^{2}_{(m,n,l-1)}\times
C_{(n,1+l+...+l^{m-1})}\times C^{2}_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(m,n,l-1,1+l+...+l^{m-1})}$,

$(ii)$ If $n$ is even, then under the assumptions of Proposition 1.2,

$\otimes^{3}G$ $\cong(A\otimes G^{ab})\times(B\otimes G)$ in which $A=\langle
X,Z\rangle$ and $B=\langle Y,T\rangle$ are subgroups of $G\otimes G$ such that
$A\otimes G^{ab}\cong C_{m}\times C_{(m,n,l-1)}\times
C_{(n,l-1,1+l+...+l^{m-1})}\times C_{(m,n,l-1,1+l+...+l^{m-1})}$, and
$B\otimes G$ is the abelian group generated by $Y_{x}=Y\otimes x,$
$Y_{y}=Y\otimes y,$ $T_{x}=T\otimes x,$ and $T_{y}=T\otimes y,$ subject to the
relations

$Y_{x}^{(m,n,2(l-1))}=Y_{y}^{(n,l-1)}=T_{y}^{(n,4(l-1),1+l+...+l^{m-1})}=1,Y_{x}^{l-1}=T_{y}^{l^{2}-1},$

and either

$\displaystyle\begin{cases}T_{x}^{(n,1+l+...+l^{m-1})}=1,&~{}~{}~{}~{}if~{}~{}~{}~{}m\equiv
0\ \textnormal{or}\ 1\ (\textnormal{mod}\ 4)\\\
T_{x}^{n}=1,T_{x}^{1+l+...+l^{m-1}}=T_{y}^{-(l-1)^{2}},&~{}~{}~{}~{}if~{}~{}~{}~{}m\equiv
2\ \textnormal{or}\ 3\ (\textnormal{mod}\ 4)\end{cases}$

when $T^{1+l+...+l^{m-1}}=1,$ or $T_{x}^{(n,1+l+...+l^{m-1})}=1$ when
$T^{1+l+...+l^{m-1}}\neq 1$,

$(iii)$ $\wedge^{3}G\cong C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(n,1+l+...+l^{m-1})}$,

$(iv)$ $\mathcal{M}^{(2)}(G)\cong C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(n,1+l+...+l^{m-1})(n,(l-1)^{2})/n}$.

It should be mentioned that the HAP package (Ellis, 2008) of GAP [13] has
methods for checking the above results in the case when $G$ is nilpotent. We
applied them for several of metacyclic groups by performing the relevant
computations (for more details see [10, Section 4]). The paper is organized as
follows. In Section 2, we study some basic properties of the tensor products
of groups including some relations between the generators of $\otimes^{3}G$.
Section 3 deals with the proof of our main theorem.

##  2 Computing The Triple Tensor Products

This section contains some results to be used throughout the rest of the
paper. We start with some familiar rules of nonabelian tensor product of
groups.

######  Proposition 2.1.

([3, Proposition 2]) Let $G$ and $H$ be two groups. Then for all $g\in G$ and
$h\in H$
$(i)$ There are homomorphism of groups $\lambda:G\otimes H\longrightarrow G$
and $\lambda^{\prime}:G\otimes H\longrightarrow H$ such that $\lambda(g\otimes
h)={g}^{h}g^{-1}$ and $\lambda^{\prime}(g\otimes h)={{}^{g}hh^{-1}}.$
$(ii)$ $\lambda(t)\otimes h=t^{h}t^{-1}$ and
$g\otimes\lambda^{\prime}(t)={{}^{g}tt^{-1}}$, and thus
$\lambda(t)\otimes\lambda^{\prime}(t_{1})=[t,t_{1}]$ for all $t,t_{1}\in
G\otimes H.$ Hence, $G$ acts trivially on $Ker\lambda^{\prime}$ and $H$ acts
trivially on $Ker\lambda.$

######  Proposition 2.2.

([3, Proposition 3]) Let $G$ and $H$ be two groups. The following relations
hold in $G\otimes H$ for all $g,g^{\prime}\in G$ and $h,h^{\prime}\in H$ (note
$[g,h]$ may be interpreted as $g^{h}g^{-1}\in G$ or ${}^{g}hh^{-1}\in H$):

$(i)~{}^{g}(g^{-1}\otimes h)=(g\otimes h)^{-1}=~{}^{h}(g\otimes h^{-1})$,

$(ii)~{}{g^{\prime}}\otimes{(^{g}hh^{-1})}={{}^{g^{\prime}}}(g\otimes
h)(g\otimes h)^{-1}$,

$(iii)~{}(g^{h}g^{-1})\otimes{h^{\prime}}=(g\otimes h)^{h^{\prime}}(g\otimes
h)^{-1}$,

$(iv)~{}(g\otimes h)(g^{\prime}\otimes h^{\prime})(g\otimes
h)^{-1}={{}^{[g,h]}}(g^{\prime}\otimes h^{\prime})$,

$(v)~{}[g\otimes
h,g^{\prime}\otimes{h^{\prime}}]=(g^{h}g^{-1})\otimes(^{g^{\prime}}{h^{\prime}}h^{\prime-1})$.

######  Corollary 2.3.

For all $a,b,c,d\in G$ the following relations hold in $G\otimes G$ and
$\otimes^{3}G$ (to simplify the notation we shall identify $b\otimes c\otimes
d$ with $(b\otimes c)\otimes d$ in $\otimes^{3}G$):

$(i)~{}^{a}(b\otimes c)=(a\otimes[b,c])(b\otimes c)$,

$(ii)~{}^{a}(b\otimes c\otimes d)=(b\otimes c\otimes d)(d\otimes[b,c]\otimes
a)$.

In next two subsections, given the metacyclic group $G$ with presentation (1),
we list some useful relations for the elements of $\otimes^{3}G$ proceeding in
a series of steps. We start with the case that $n$ is odd which is relatively
straightforward. Note by Proposition 2.2 that $[a\otimes b\otimes
c,a^{\prime}\otimes b^{\prime}\otimes c^{\prime}]=[[a,b]\otimes
c,[a^{\prime},b^{\prime},c^{\prime}]]$, for all
$a,b,c,a^{\prime},b^{\prime},c^{\prime}\in G$. Now by setting $G=G\otimes G$
and $H=G$ in Proposition 2.1, it tells us immediately that
$Ker(\lambda^{\prime}:\otimes^{3}G\rightarrow G)$ is central. Since
$Im\lambda^{\prime}=\gamma_{3}(G)$ is cyclic, it follows that $\otimes^{3}G$
is abelian.

###  2.1 The n odd case

Throughout this section we let $G$ be an arbitrary finite split metacyclic
group as in (1) and $n$ is odd.

$\displaystyle{{}^{y}}(x\otimes y\otimes y)={{}^{x}}(x\otimes y\otimes
y)=x\otimes y\otimes y.$ (2)

###### Proof.

Taking $G=G\otimes G$ and $H=G$ in Proposition 2.1 we get the homomorphism
$\lambda:\otimes^{3}G\rightarrow G\otimes G$ so that $\lambda(a\otimes
b\otimes c)=(a\otimes b)^{c}(a\otimes b)^{-1}$, for all $a,b,c\in G$. In
addition, it follows from the Proposition 2.2$(iii)$ and [3, (5.1)] that
$x\otimes y\otimes y\in Ker\lambda$. Thus Proposition 2.1 concludes the
result. ∎

$\displaystyle(x\otimes y\otimes y)^{l-1}=1.$ (3)

###### Proof.

It follows from [3, (5.3)] that $(x\otimes y){{}^{x}}(x\otimes
y)^{-1}=(x\otimes y)^{1-l}$. According to the defining actions in
$\otimes^{3}G$, as ${}^{x\otimes
y}xx^{-1}=[x,y,x]=[y^{l-1},x]=y^{-(l-1)^{2}},$ then

$\displaystyle(x\otimes y\otimes y)^{(l-1)^{3}}$ $\displaystyle=(x\otimes
y)^{1-l}\otimes y^{-(l-1)^{2}}$ $\displaystyle=(x\otimes y)\ ^{x}(x\otimes
y)^{-1}\otimes{{}^{(x\otimes y)}x}x^{-1}$ $\displaystyle=[x\otimes y\otimes
x,x\otimes y\otimes x]=1.$

On the other hand from (2) and [3, (5.3)] we have

$\displaystyle(x\otimes y\otimes y)$ $\displaystyle={{}^{x}(x\otimes y\otimes
y)}={{}^{x}(x\otimes y)}\otimes{{}^{x}y}=(x\otimes y)^{l}\otimes
y^{l}=(x\otimes y\otimes y)^{l^{2}},$

which implies that $(x\otimes y\otimes y)^{l^{2}-1}=1$. Hence

$\ 1=(x\otimes y\otimes y)^{(l-1)^{3}-(l^{2}-1)}=(x\otimes y\otimes
y)^{l^{3}-4l^{2}+3l}=(x\otimes y\otimes y)^{4(l-1)}.$

Now since $n$ is odd, it finishes the proof. ∎

${}^{y}(x\otimes y\otimes x)=x\otimes y\otimes x.$ (4)

###### Proof.

From [3, (5.3)] and (3) we have that

${}^{y}(x\otimes y\otimes x)$ $\displaystyle=((x\otimes y)\otimes{{}^{y}x)}$
$\displaystyle=((x\otimes y)\otimes y^{1-l}x)$ $\displaystyle=(x\otimes
y\otimes y^{1-l}){{}^{y^{1-l}}(x\otimes y\otimes x)}$ $\displaystyle=(x\otimes
y\otimes y)^{1-l}\ {}^{y^{1-l}}(x\otimes y\otimes x)$
$\displaystyle={{}^{y^{1-l}}(x\otimes y\otimes x)},$

which implies $x\otimes y\otimes x$ is fixed by $y^{l}.$ In addition, as $l$
and $n$ are relatively prime, there exist integers $s,t$ such that $sl+tn=1$.
Hence ${{}^{y}}(x\otimes y\otimes x)={{}^{y^{sl+tn}}(x\otimes y\otimes
x)}={{}^{y^{sl}}(x\otimes y\otimes x)}=x\otimes y\otimes x.$ ∎

${}^{x}(x\otimes y\otimes x)=(x\otimes y\otimes x)^{l}.$ (5)

###### Proof.

By applying [3, (5.3)] we see that ${}^{x}(x\otimes y\otimes x)=(x\otimes
y)^{l}\otimes x$. Now the result follows by induction on $l$ together with
using (4). ∎

Setting $i.j=i(1+l+l^{2}+...+l^{j-1})$, then

$(x\otimes y)^{i}\otimes x^{j}=(x\otimes y\otimes x)^{i.j}$ and $(x\otimes
y)^{i}\otimes y^{k}=(x\otimes y\otimes y)^{ik}$, for any integers $i,j,k$. (6)

###### Proof.

It follows by applying [3, (5.2)], (2), (4), and (5). ∎

###  2.2 The n even case

In this subsection we are concerned with the case that $G$ is a finite split
metacyclic group as in (1) and $n$ is even. As we’ll see, this case is
slightly more complicated with respect to the odd case. For instance, the
order of $y\otimes y\otimes y$ would be discussed here while in odd case it is
easily determined as a direct factor of the abelian group $\nabla(G)\otimes
G^{ab}$ (see Section 3 for more details).

$\displaystyle(y\otimes y\otimes y)^{l-1}=1.$ (7)

###### Proof.

It is readily seen that ${{}^{x}}(y\otimes y\otimes y)=y\otimes y\otimes
y^{l}=(y\otimes y\otimes y)^{l}$. In addition, since $y\otimes y\otimes y$
belongs to $Ker\lambda$ given in the proof of (2), it follows that
${{}^{x}}(y\otimes y\otimes y)=y\otimes y\otimes y$. The result now follows
easily. ∎

$\displaystyle{{}^{y}}(x\otimes y\otimes y)=x\otimes y\otimes y.$ (8)

###### Proof.

Using [11, (9)] and (7),

${}^{y}(x\otimes y\otimes y)$ $\displaystyle={{}^{y}(x\otimes y)\otimes y}$
$\displaystyle=(x\otimes y)(y\otimes y)^{l-1}\otimes y$
$\displaystyle={{}^{x\otimes y}((y\otimes y)^{l-1}\otimes y)(x\otimes y\otimes
y)}$ $\displaystyle=(x\otimes y\otimes y)(y\otimes y\otimes y)^{l-1}$
$\displaystyle=x\otimes y\otimes y.$

∎

$(x\otimes y)^{i}\otimes y^{j}=(x\otimes y\otimes y)^{ij}$, for any integers
$i,j$. (9)

###### Proof.

It can be proved by induction on $i$ together with using (8). ∎

$\displaystyle{{}^{x}}(x\otimes y\otimes y)=(x\otimes y\otimes y)^{l^{2}}.$
(10)

###### Proof.

Invoking [11, (10)], (8) and (9),

$\displaystyle{{}^{x}}(x\otimes y\otimes y)$ $\displaystyle={{}^{x}(x\otimes
y)\otimes{{}^{x}y}}$ $\displaystyle=(x\otimes y)^{l}(y\otimes
y)^{-l(l-1)^{2}/2}\otimes y^{l}$ $\displaystyle={{}^{(x\otimes
y)^{l}}}((y\otimes y)^{-l(l-1)^{2}/2}\otimes y^{l})((x\otimes y)^{l}\otimes
y^{l})$ $\displaystyle=(y\otimes y\otimes y)^{-l^{2}(l-1)^{2}/2}(x\otimes
y\otimes y)^{l^{2}}.$

The result follows by (7). ∎

$\displaystyle(x\otimes y\otimes y)^{1+l+...+l^{m-1}}=1.$ (11)

###### Proof.

Clearly the assertion holds when $(x\otimes y)^{1+l+...+l^{m-1}}=1.$ If
$(x\otimes y)^{1+l+...+l^{m-1}}\neq 1$, then by Proposition 1.2 as $A^{2}=1$
we see $(x\otimes y)^{1+l+...+l^{m-1}}=(y\otimes y)^{l-1}$. Hence $(x\otimes
y)^{1+l+...+l^{m-1}}\otimes y=(y\otimes y)^{l-1}\otimes y$, and consequently
(7) concludes the result. ∎

$\displaystyle(x\otimes y\otimes y)^{4(l-1)}=1.$ (12)

###### Proof.

As $(y\otimes y)^{2(l-1)}=1$ by [11, (15)], then it follows that $y$ fixes
$(x\otimes y)^{2}$. So the element $(x\otimes y\otimes y)^{2}$ = $(x\otimes
y)^{2}\otimes y$ belongs to $Ker\lambda,$ where $\lambda$ is the homomorphism
given in the proof of (2). Now Proposition 2.1$(ii)$ implies the group $G$
acts on $(x\otimes y\otimes y)^{2}$ trivially. Consequently it follows by (10)
that $(x\otimes y\otimes y)^{2}={{{}^{x}}(x\otimes y\otimes y)^{2}}=(x\otimes
y\otimes y)^{2l^{2}}.$ Therefore $(x\otimes y\otimes y)^{2(l^{2}-1)}=1$. On
the other hand, from [11, (10)] and Proposition 2.2 we have $(x\otimes
y)^{x}(x\otimes y)^{-1}$ =$(x\otimes y)^{1-l}(y\otimes y)^{(l-1)^{2}/2}$. Thus
$(x\otimes y)^{1-l}=(x\otimes y)^{x}(x\otimes y)^{-1}(y\otimes
y)^{-(l-1)^{2}/2}$. Hence by applying (7) we have

$\displaystyle(x\otimes y\otimes y)^{(l-1)^{3}}$ $\displaystyle=(x\otimes
y)^{1-l}\otimes y^{-(l-1)^{2}}$ $\displaystyle=((x\otimes y)^{x}(x\otimes
y)^{-1}(y\otimes y)^{-(l-1)^{2}/2})\otimes y^{-(l-1)^{2}}$
$\displaystyle=((y\otimes y)^{-(l-1)^{2}/2}\otimes y^{-(l-1)^{2}})((x\otimes
y)^{x}(x\otimes y)^{-1}\otimes y^{-(l-1)^{2}})$ $\displaystyle=(y\otimes
y\otimes y)^{{(l-1)^{2}\over{2}}-(l-1)^{2}}((x\otimes y)^{x}(x\otimes
y)^{-1}\otimes{{}^{x\otimes y}x}x^{-1})$ $\displaystyle=[x\otimes y\otimes x,\
x\otimes y\otimes x]=1.$

Combining two last equations leads to the assertion. ∎

$\displaystyle{{}^{y}}(x\otimes y\otimes x)=(x\otimes y\otimes x)(x\otimes
y\otimes y)^{l-1}.$ (13)

###### Proof.

Using Corollary 2.3, [11, (11)] and (7),

$\displaystyle{{}^{y}}(x\otimes y\otimes x)$
$\displaystyle=((x\otimes[x,y]\otimes y)(x\otimes y\otimes x)$
$\displaystyle=[(x\otimes y)^{l-1}(y\otimes y)^{-(l-1)^{2}(l-2)/2}\otimes
y](x\otimes y\otimes x)$ $\displaystyle=(x\otimes y\otimes x)(x\otimes
y\otimes y)^{l-1}.$

∎

$(x\otimes y)^{i}\otimes x=(x\otimes y\otimes x)^{i}(x\otimes y\otimes
y)^{{i\choose 2}(l-1)^{2}}$, for any integer $i$. (14)

###### Proof.

It follows by induction on $i$, [3, (22)], (9), and (13). ∎

$\displaystyle(x\otimes y\otimes y)^{l^{2}-1}=(y\otimes y\otimes x)^{l-1}.$
(15)

###### Proof.

Use (10) and Corollary 2.3,

$(x\otimes y\otimes y)^{l^{2}}={{}^{x}}(x\otimes y\otimes
y)=(y\otimes[x,y]\otimes x)(x\otimes y\otimes y)=(y\otimes y\otimes
x)^{l-1}(x\otimes y\otimes y).$

∎

$\displaystyle{{}^{x}}(x\otimes y\otimes x)=(x\otimes y\otimes x)^{l}(x\otimes
y\otimes y)^{(l-1)^{2}}.$ (16)

###### Proof.

First note that $G$ acts trivially on $y\otimes y\otimes x$. It follows by
Corollary 2.3, [11, (11)], (14), and (15) that

$\displaystyle{{}^{x}}(x\otimes y\otimes x)$
$\displaystyle=(x\otimes[x,y]\otimes x)(x\otimes y\otimes x)$
$\displaystyle=[(x\otimes y)^{l-1}(y\otimes y)^{-(l-1)^{2}(l-2)/2}\otimes
x](x\otimes y\otimes x)$ $\displaystyle=(y\otimes y\otimes
x)^{-(l-1)^{2}(l-2)/2}((x\otimes y)^{l-1}\otimes x)(x\otimes y\otimes x)$
$\displaystyle=(y\otimes y\otimes x)^{-(l-1)^{2}(l-2)/2}(x\otimes y\otimes
x)^{l-1}(x\otimes y\otimes y)^{(l-2)(l-1)^{3}/2}(x\otimes y\otimes x)$
$\displaystyle=(x\otimes y\otimes x)^{l}(x\otimes y\otimes
y)^{-(l-2)(l-1)^{2}}.$

So the result is obviously true by applying the last equation used in the
proof of (12). ∎

For any integers $i$ and $j$ we have:

$\displaystyle(x\otimes y)^{i}\otimes
x^{j}=\left\\{\begin{array}[]{lll}(x\otimes y\otimes x)^{i.j}\ (x\otimes
y\otimes y)^{{i\choose 2}.j(l-1)^{2}}&~{}~{}~{}~{}if~{}~{}~{}~{}j\equiv 0\
\textnormal{or}\ 1\ (\textnormal{mod}\ 4)\\\ (x\otimes y\otimes x)^{i.j}\
(x\otimes y\otimes y)^{({i\choose
2}.j+i)(l-1)^{2}}&~{}~{}~{}~{}if~{}~{}~{}~{}j\equiv 2\ \textnormal{or}\ 3\
(\textnormal{mod}\ 4)\end{array}\right.$

where $i.j=i(1+l+l^{2}+...+l^{j-1})$.

###### Proof.

We proceed by induction on both $i$ and $j$. Assuming $i=1$, it follows by
induction on $j$ together with (16) that

$\displaystyle x\otimes y\otimes x^{j}=\left\\{\begin{array}[]{lll}(x\otimes
y\otimes x)^{1.j}&~{}~{}~{}~{}if~{}~{}~{}~{}j\equiv 0\ \textnormal{or}\ 1\
(\textnormal{mod}\ 4)\\\ (x\otimes y\otimes x)^{1.j}\ (x\otimes y\otimes
y)^{(l-1)^{2}}&~{}~{}~{}~{}if~{}~{}~{}~{}j\equiv 2\ \textnormal{or}\ 3\
(\textnormal{mod}\ 4)\end{array}\right.$

It suffices to prove the second part of the assertion and the first is
similar. Using (13), $l-1$ times, we have

$\displaystyle(x\otimes y)^{i+1}\otimes x^{j}$ $\displaystyle={{}^{x\otimes
y}}((x\otimes y)^{i}\otimes x^{j})(x\otimes y\otimes x^{j})$
$\displaystyle={{}^{x\otimes y}}((x\otimes y\otimes x)^{i.j}(x\otimes y\otimes
y)^{({i\choose 2}.j+i)(l-1)^{2}})(x\otimes y\otimes x)^{1.j}(x\otimes y\otimes
y)^{(l-1)^{2}}$ $\displaystyle=(x\otimes y\otimes x)^{i.j}(x\otimes y\otimes
y)^{i.j(l-1)^{2}}(x\otimes y\otimes y)^{({i\choose 2}.j+i)(l-1)^{2}}(x\otimes
y\otimes x)^{1.j}(x\otimes y\otimes y)^{(l-1)^{2}}$ $\displaystyle=(x\otimes
y\otimes x)^{(i+1).j}(x\otimes y\otimes y)^{({{i+1}\choose
2}.j+i+1)(l-1)^{2}}.$

∎

##  3 Proof of The Main Theorem

In this section we prove the main theorem of the paper. The proof relies on
the previous section together with a computation method based on the crossed
pairings. There is a significant difference between the even and odd case, and
w’ll treat them separately beginning with the odd case.

###  3.1 The n odd case

Let $G$ be a finite split metacyclic group as in (1) and $n$ is odd. We
observe from Proposition 1.1 that in this case $G\otimes G$ splits as

$G\otimes G\cong\nabla(G)\times(G\wedge G)\cong C_{m}\times C_{(n,l-1)}\times
C_{(n,l-1,1+l+...+l^{m-1})}\times C_{(n,1+l+...+l^{m-1})},$

where $G\wedge G$ is isomorphic to the subgroup of $G\otimes G$ generated by
$\langle x\otimes y\rangle$. This is not the case in general when $n$ is even.
The groups $G\wedge G$ and $G$ act on each other in such a way that $G\wedge
G\cong\langle x\otimes y\rangle$ is fixed under the action of $G$. So by
applying [3, Proposition 10] one could describe $\otimes^{3}G$ as follows:

$\ \otimes^{3}G\cong(\nabla(G)\times(G\wedge G))\otimes
G\cong(\nabla(G)\otimes G)\times(G\wedge G\otimes G).$

Now since the groups $\nabla(G)$ and $G$ act on each other trivially,
Proposition 2.4 in [4] allows us to express $\nabla(G)\otimes G$ as the tensor
product $\nabla(G)\otimes G^{ab}$ of abelian groups. Therefore by Proposition
1.1 we conclude that

$\nabla(G)\otimes G^{ab}\cong C_{m}\times C_{(n,l-1)}\times
C^{2}_{(m,n,l-1)}\times C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(m,n,l-1,1+l+...+l^{m-1})}.$

The main task then is to determine the cyclic invariants of $G\wedge G\otimes
G$. For this purpose first we expand the arbitrary element $(x\otimes
y)^{i}\otimes$ $y^{j}x^{k}$ of $G\wedge G\otimes G$ by invoking (4) and (6):

$\displaystyle(x\otimes y)^{i}\otimes y^{j}x^{k}$ $\displaystyle=((x\otimes
y)^{i}\otimes y^{j})\ {{}^{y^{j}}}((x\otimes y)^{i}\otimes x^{k})$
$\displaystyle=(x\otimes y\otimes y)^{ij}\ {{}^{y^{j}}}(x\otimes y\otimes
x)^{i.k}$ $\displaystyle=(x\otimes y\otimes y)^{ij}(x\otimes y\otimes
x)^{i.k}.$

This establishes what the generators of $G\wedge G\otimes G$ are. To find the
orders of these generators, and any possible relations among them, by
considering our analysis of $\otimes^{3}G$ in Subsection 2.1, we use crossed
pairing into suitable cyclic groups. Recall that for groups $G,H$ and $L$
where $G$ and $H$ acting upon each other compatibly and acting upon themselves
by conjugation, a function $\Phi:G\times H\longrightarrow L$ is called a
crossed pairing if

$\Phi(gg^{\prime},h)=\Phi(^{g}g^{\prime},^{g}h)\Phi(g,h)\hskip
14.22636pt\textnormal{and}\hskip
14.22636pt\Phi(g,hh^{\prime})=\Phi(g,h)\Phi(^{h}g,^{h}h^{\prime}),$

for all $g,g^{\prime}\in G$, $h,h^{\prime}\in H$. Clearly any crossed pairing
$\Phi:G\times H\longrightarrow L$ determines a unique homomorphism
$\Phi^{*}:G\otimes H\longrightarrow L$ such that $\Phi^{*}(g\otimes
h)=\Phi(g,h).$

Now define $\Phi:(G\wedge G)\times G\rightarrow C_{(n,1+l+...+l^{m-1})}\times
C_{(n,l-1,1+l+...+l^{m-1})}$ by $\Phi((x\otimes y)^{i},y^{j}x^{k})=a^{i.k}\
b^{ij}$ and let $((x\otimes y)^{i},y^{j}x^{k})=((x\otimes
y)^{i^{\prime}},y^{j^{\prime}}x^{k^{\prime}})$. Then with setting
$r=(n,1+l+...+l^{m-1})$, it follows that $i^{\prime}=i+ur,j^{\prime}=j+vn$,
and $k^{\prime}=k+wm,$ for some integers $u,v,w$. So
$a^{i^{\prime}k^{\prime}}b^{i^{\prime}j^{\prime}}=a^{i.k}\ b^{ij}$, which
implies that $\Phi$ is well-defined. We prove that $\Phi$ is a crossed
pairing. The following verifies the first rule for a crossed pairing:

$\displaystyle\Phi(^{(x\otimes y)^{i}}(x\otimes y)^{i^{\prime}},\ ^{(x\otimes
y)^{i}}y^{j}x^{k})\Phi((x\otimes y)^{i},\ y^{j}x^{k})$
$\displaystyle=\Phi((x\otimes y)^{i^{\prime}},\
y^{j+i(l-1)(1-l^{k})}x^{k})\Phi((x\otimes y)^{i},\ y^{j}x^{k})$
$\displaystyle=a^{i^{\prime}.k}b^{i^{\prime}j+i^{\prime}i(l-1)(1-l^{k})}a^{i.k}\
b^{ij}$ $\displaystyle=a^{(i+i^{\prime}).k}\ b^{(i+i^{\prime})j}$
$\displaystyle=\Phi((x\otimes y)^{i}(x\otimes y)^{i^{\prime}}\otimes
y^{j}x^{k}).$

The other rule follows by [3, (22), (5.2), (5.3)]:

$\displaystyle\Phi((x\otimes y)^{i},y^{j}x^{k})\Phi(^{y^{j}x^{k}}(x\otimes
y)^{i},^{y^{j}x^{k}}y^{j^{\prime}}x^{k^{\prime}})$
$\displaystyle=\Phi((x\otimes y)^{i},\ y^{j}x^{k})\Phi((x\otimes
y)^{il^{k}},y^{j^{\prime}l^{k}+j(1-l^{k^{\prime}})}x^{k^{\prime}})$
$\displaystyle=a^{i.k}b^{ij}a^{il^{k}.k^{\prime}}\
b^{il^{k}(j^{\prime}l^{k}+j(1-l^{k^{\prime}}))}$
$\displaystyle=a^{i(1+l+...+l^{k-1})+i(l^{k}+l^{k+1}+...+l^{k+k^{\prime}-1})}\
b^{i(j+j^{\prime}l^{k})}$ $\displaystyle=a^{i.(k+k^{\prime})}\
b^{i(j+j^{\prime}l^{k})}$ $\displaystyle=\Phi((x\otimes y)^{i},\
y^{j+j^{\prime}l^{k}}x^{k+k^{\prime}})$ $\displaystyle=\Phi((x\otimes y)^{i},\
y^{j}x^{k}y^{j^{\prime}}x^{k^{\prime}}).$

Thus $\Phi$ induces a homomorphism ${\Phi^{\ast}}:G\wedge G\otimes
G\rightarrow C_{(n,1+l+...+l^{m-1})}$ $\times C_{(n,l-1,1+l+...+l^{m-1})},$
which shows that the generators $x\otimes y\otimes x$ and $x\otimes y\otimes
y$ are independent and have the orders divided by $(n,1+l+...+l^{m-1})$ and
$(n,l-1,1+l+...+l^{m-1})$, respectively. On the other hand we simply note by
Proposition 1.1 and (6) that $(x\otimes y\otimes x)^{(n,1+l+...+l^{m-1})}=1$.
Likewise it follows from Proposition 1.1, (2) and (3) that $(x\otimes y\otimes
y)^{(n,l-1,1+l+...+l^{m-1})}=1$. Hence $|x\otimes y\otimes
x|=(n,1+l+...+l^{m-1})$ and $|x\otimes y\otimes y|=(n,l-1,1+l+...+l^{m-1})$;
from which we conclude that

$G\wedge G\otimes G\cong C_{(n,1+l+...+l^{m-1})}\times
C_{(n,l-1,1+l+...+l^{m-1})},$

as desired. Consequently as $x\otimes y\otimes[x,y]=x\otimes y\otimes
y^{l-1}=(x\otimes y\otimes y)^{l-1}=1$ by (2) and (3), it follows by
definition that $\wedge^{3}G\cong\ G\wedge G\otimes G.$

Now we are ready to complete the proof of the main theorem in odd case by
computing $\mathcal{M}^{(2)}(G)$. Our method relies on a tensor product
approach due to Burns and Ellis [5]. Let $\gamma_{3}^{\sharp}(G)$ be the
quotient group $\wedge^{3}G/\tau(G)$ where $\tau(G)$ is the normal subgroup of
$\wedge^{3}G$ generated by the elements $\langle a,b,c\rangle=((a\wedge
b)\wedge{{}^{b}}c)((b\wedge c)\wedge{{}^{c}}a)((c\wedge a)\wedge{{}^{a}}b),$
for all $a,b,c\in G$. By the Hall–Witt commutator identity, the homomorphism
$[~{},~{},~{}]:\wedge^{3}G\longrightarrow G$ induces a homomorphism
$[~{},~{},~{}]:\gamma_{3}^{\sharp}(G)\longrightarrow G$. It is well-known [5,
Theorem 2.9] that the surjection
$\ker([~{},~{},~{}]:\wedge^{3}G\longrightarrow
G)\twoheadrightarrow\mathcal{M}^{(2)}(G)$ given in [5, Theorem 2.6] gives rise
to the natural isomorphism

$\displaystyle\mathcal{M}^{(2)}(G)\cong\ker([~{},~{},~{}]:\gamma_{3}^{\sharp}(G)\longrightarrow
G).$ (18)

So in order to describe $\mathcal{M}^{(2)}(G),$ first, it is required to
evaluate the subgroup $\tau(G)$ of $\wedge^{3}G.$ By invoking (2) and (4)
together with the initial relations of $G\wedge G$ we have

$\displaystyle\langle x,y,y\rangle$ $\displaystyle=(x\wedge
y\wedge{{}^{y}y)}(y\wedge y\wedge{{}^{y}x)}(y\wedge x\wedge{{}^{x}y)}$
$\displaystyle=(x\wedge y\wedge y)(y\wedge x\wedge y^{l})$
$\displaystyle=(x\wedge y\wedge y)((x\wedge y)^{-1}\wedge y)^{l}$
$\displaystyle=(x\wedge y\wedge y)(x\wedge y\wedge y)^{-l}$
$\displaystyle=(x\wedge y\wedge y)^{1-l}=1.$

Analogously (2), (3) and (4) imply that

$\displaystyle\langle x,y,x\rangle$ $\displaystyle=(x\wedge
y\wedge{{}^{y}x)}(y\wedge x\wedge{{}^{x}x)}(x\wedge x\wedge{{}^{x}y)}$
$\displaystyle=(x\wedge y\wedge y^{1-l}x)((x\wedge y)^{-1}\wedge x)$
$\displaystyle=(x\wedge y\wedge y^{1-l})\ {{}^{y^{1-l}}(x\wedge y\wedge
x)(x\wedge y\wedge x)^{-1}}$ $\displaystyle=(x\wedge y\wedge y)^{1-l}=1.$

Hence $\tau(G)$ is trivial and consequently $\gamma_{3}^{\sharp}(G)$
$\cong\wedge^{3}G.$ As $\gamma_{3}(G)$ = $\langle y^{(l-1)^{2}}\rangle,$ then
it readily follows by (18) that

$\displaystyle\mathcal{M}^{(2)}(G)\cong C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(n,1+l+...+l^{m-1})(n,(l-1)^{2})/n}.$

###  3.2 The n even case

As can be expected, the $n$ even case presents more difficulties in
computation of $\otimes^{3}G$; here Proposition 1.2 does not give us a
splitting of $G\otimes G$ into $\nabla(G)\times(G\wedge G)$, the orders of
generators are not completely determined by the analysis in Subsection 2.2,
and the identities found there are a bit more complex (e.g., compare the $n$
odd case with the $n$ even case in (6) and (17)). The reason is that for the
even case the subgroup $\langle x\otimes y\rangle$ is not fixed under the
action of $G$ (see [11, (9) and (10)]).

So in order to satisfy the hypotheses of [3, Proposition 10] we let $G\otimes
G\cong$ $A\times B$, where $A=\langle X,Z\rangle$ and $B=\langle Y,T\rangle$
are the subgroups of $G\otimes G$ constructed by using Proposition 1.2. Thus
like the discussion at the beginning of Subsection 3.1 it follows that

$\displaystyle\otimes^{3}G$ $\displaystyle\cong(A\otimes G)\times(B\otimes
G)\cong C_{m}\times C_{(m,n,l-1)}\times C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(m,n,l-1,1+l+...+l^{m-1})}\times(B\otimes G),$

whence it is enough to work on the cyclic invariants of the subgroup $B\otimes
G.$

Assume $Y^{i}T^{j}\in B$ and $y^{s}x^{t}\in G$ for some integers $i,j,s,t,$
and put $Y_{x}=Y\otimes x,Y_{y}=Y\otimes y,T_{x}=T\otimes x,T_{y}=T\otimes y.$
From (8), (9), (13), (17), and the argument in the proof of (2) which shows
that $Y_{x}$ is fixed under the action of $G$, we have that

$\displaystyle Y^{i}T^{j}\otimes y^{s}x^{t}$ $\displaystyle=(Y^{i}T^{j}\otimes
y^{s})\ ^{y^{s}}(Y^{i}T^{j}\otimes x^{t})$ $\displaystyle=(T^{j}\otimes
y^{s})(Y^{i}\otimes y^{s})\ {{}^{y^{s}}}(T^{j}\otimes x^{t})\
{{}^{y^{s}}}(Y^{i}\otimes x^{t})$ $\displaystyle=T_{y}^{js}\ Y_{y}^{is}\
{{}^{y^{s}}}(T_{x}^{j.t}\ T_{y}^{k})\ {{}^{y^{s}}}Y_{x}^{it}$
$\displaystyle=T_{y}^{js}\ Y_{y}^{is}\ (T_{x}^{j.t}\ T_{y}^{k+s(l-1)j.t})\
Y_{x}^{it}$ $\displaystyle=Y_{x}^{it}\ Y_{y}^{is}\ T_{x}^{j.t}\
T_{y}^{k+s(l-1)j.t+js},$

where $k$ is the correspondence power of $T_{y}$ which already occurred in
(17). This tells us that $B\otimes G$ is generated by the four elements
$Y_{x},Y_{y},T_{x},$ and $T_{y}$.

It remains to determine the relations among these generators. As noted above,
$Y_{x}$ is fixed under the action of $G$. From Proposition 1.2, as
$Y^{n}=Y^{2(l-1)}=1$, it follows that $Y_{x}^{n}=Y_{x}^{2(l-1)}=1$. Also we
have $1=Y\otimes x^{m}=(Y\otimes x)^{m}$. So $Y_{x}^{(m,n,2(l-1))}=1$.
Similarly and by (7) we obtain $Y_{y}^{(n,l-1)}=1$. Since $T^{n}=1$, it is
readily seen by (8) that $T_{y}^{n}=1$. This together with (11) and (12)
conclude that $T_{y}^{(n,4(l-1),1+l+...+l^{m-1})}=1$. Furthermore, (15) gives
us $Y_{x}^{l-1}=T_{y}^{l^{2}-1}$. Now, as the relation
$T^{1+l+...+l^{m-1}}=A^{(l-1)(m-1)m/4}$ in Proposition 1.2 affects our
discussion, we proceed with the following two cases:
Case 1) If $T^{1+l+...+l^{m-1}}=1$.

Put $(i,j)=(1,m)$ in (17). It gives either $T_{x}^{1+l+...+l^{m-1}}=1$ or
$T_{x}^{1+l+...+l^{m-1}}=T_{y}^{-(l-1)^{2}}$. On the other hand, taking
$(i,j)=(n,1)$ in (17) it implies either $T_{x}^{n}=T_{y}^{-{n\choose
2}(l-1)^{2}}$ or $T_{x}^{n}=T_{y}^{-({n\choose 2}+n)(l-1)^{2}}$. By the fact
$T_{y}^{n}=1$ it is now trivial that $T_{x}^{n}=1$. So we obtain the last
relation equivalent to

$\displaystyle\begin{cases}T_{x}^{(n,1+l+...+l^{m-1})}=1,&~{}~{}~{}~{}if~{}~{}~{}~{}m\equiv
0\ \textnormal{or}\ 1\ (\textnormal{mod}\ 4)\\\
T_{x}^{n}=1,T_{x}^{1+l+...+l^{m-1}}=T_{y}^{-(l-1)^{2}},&~{}~{}~{}~{}if~{}~{}~{}~{}m\equiv
2\ \textnormal{or}\ 3\ (\textnormal{mod}\ 4)\end{cases}$

Case 2) If $T^{1+l+...+l^{m-1}}\neq 1$.

If $l-1$ is divisible by 4, then clearly $(l-1)(m-1)m/4$ is even. When $l-1$
isn’t divisible by 4 and $m\equiv 0\ \textnormal{or}\ 1\ (\textnormal{mod}\
4)$, then again $(l-1)(m-1)m/4$ is even. Hence
$T^{1+l+...+l^{m-1}}=A^{(l-1)(m-1)m/4}=1$ and we are reduced to the first
case. So it only remains the case when $l-1$ isn’t divisible by 4 and $m\equiv
2\ \textnormal{or}\ 3\ (\textnormal{mod}\ 4)$. It is obvious that
$(l-1)(m-1)m/4$ is odd, whence $T^{1+l+...+l^{m-1}}=Y^{l-1}$. By applying (14)
we see

$\displaystyle Y_{x}^{l-1}=Y^{l-1}\otimes x=T^{1+l+...+l^{m-1}}\otimes
x=T_{x}^{1+l+...+l^{m-1}}T_{y}^{{1+l+...+l^{m-1}\choose 2}(l-1)^{2}}.$

As $(m,n,2(l-1))$ divides $l-1,$ then $Y_{x}^{l-1}=1$. Therefore it follows by
(11) that $T_{x}^{1+l+...+l^{m-1}}=1.$ Consequently from the argument in the
first case we deduce that $T_{x}^{(n,1+l+...+l^{m-1})}=1$.

Now by imposing the relation $x\otimes y\otimes[x,y]=x\otimes y\otimes
y^{l-1}=(x\otimes y\otimes y)^{l-1}=1$ to $G\wedge G\otimes G=\langle
T_{x},~{}T_{y}\rangle$, it is readily obtained that

$\wedge^{3}G\cong C_{(n,1+l+..+l^{m-1})}\times C_{(n,l-1,1+l+...+l^{m-1})}.$

Finally, we evaluate the subgroup $\tau(G)$ of $\wedge^{3}G.$ Likewise the odd
case at the end of Subsection 3.1, we first observe that $\langle
x,y,y\rangle=1$. Also by invoking (13) we get

$\displaystyle\langle x,y,x\rangle$ $\displaystyle=(x\wedge
y\wedge{{}^{y}x)}(y\wedge x\wedge{{}^{x}x)}(x\wedge x\wedge{{}^{x}y)}$
$\displaystyle=(x\wedge y\wedge y^{1-l}x)((x\wedge y)^{-1}\wedge x)$
$\displaystyle=(x\wedge y\wedge y^{1-l})\ {{}^{y^{1-l}}(x\wedge y\wedge
x)(x\wedge y\wedge x)^{-1}}$ $\displaystyle=(x\wedge y\wedge y)^{l(1-l)}=1.$

As a consequence $\tau G)\cong\langle 1\rangle$ and then
$\gamma_{3}^{\sharp}(G)$ $\cong\wedge^{3}G.$ As before, the isomorphism (18)
now implies that

$\displaystyle\mathcal{M}^{(2)}(G)\cong C_{(n,l-1,1+l+...+l^{m-1})}\times
C_{(n,1+l+...+l^{m-1})(n,(l-1)^{2})/n}.$

## Statements and Declarations

Competing interests: On behalf of all authors, the corresponding author states
that there is no conflict of interest.

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E-mail address<EMAIL_ADDRESS>s.hadi<EMAIL_ADDRESS>


# Neuroevolutionary learning of particles and protocols for self-assembly

Stephen Whitelam<EMAIL_ADDRESS>Molecular Foundry, Lawrence Berkeley
National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA  Isaac Tamblyn
<EMAIL_ADDRESS>National Research Council of Canada, Ottawa, ON, Canada
Vector Institute for Artificial Intelligence, Toronto, ON, Canada

###### Abstract

Within simulations of molecules deposited on a surface we show that
neuroevolutionary learning can design particles and time-dependent protocols
to promote self-assembly, without input from physical concepts such as thermal
equilibrium or mechanical stability and without prior knowledge of candidate
or competing structures. The learning algorithm is capable of both directed
and exploratory design: it can assemble a material with a user-defined
property, or search for novelty in the space of specified order parameters. In
the latter mode it explores the space of what can be made rather than the
space of structures that are low in energy but not necessarily kinetically
accessible.

Introduction— How do we make a material with specified properties? In pursuit
of “synthesis by design” De Yoreo  _et al._ (2016); Broholm  _et al._ (2016)
the materials science community has developed and adapted algorithms of
inverse design and machine learning. These approaches can identify
interparticle potentials able to stabilize target structures or promote their
self-assembly from solution Cohn and Kumar (2009); Torquato (2009); Bianchi
_et al._ (2012); Rechtsman  _et al._ (2006); Lindquist  _et al._ (2016);
Jadrich  _et al._ (2017); Long and Ferguson (2018); Gadelrab  _et al._ (2017);
Ferguson (2017); Piñeros  _et al._ (2018); van Anders  _et al._ (2015); Adorf
_et al._ (2018); Madge and Miller (2017); Jiang  _et al._ (2018); Kumar  _et
al._ (2019); Zhou  _et al._ (2019); Sherman  _et al._ (2020); Reinhardt and
Frenkel (2014); Romano  _et al._ (2020); Miskin  _et al._ (2016), and can
identify protocols or reaction conditions that optimize the self-assembly of
specified particles Klotsa and Jack (2013); Miskin  _et al._ (2016); Raccuglia
_et al._ (2016); Whitelam and Tamblyn (2020); Tang  _et al._ (2016).

Here we present an approach based on evolutionary learning Holland (1992) that
simultaneously designs particles and protocols in order to self-assemble
materials to order. We study a coarse-grained computational model of molecular
self-assembly at a surface Bartels (2010); Elemans  _et al._ (2009); Swiegers
and Malefetse (2002). Coarse-grained models are simple by design Doye  _et
al._ (2004); Hagan and Chandler (2006); Molinero and Moore (2008); Romano and
Sciortino (2011); Glotzer  _et al._ (2004); Doye  _et al._ (2007); Rapaport
(2010); Murugan  _et al._ (2015); Whitelam and Jack (2015); Grunwald and
Geissler (2014); Nguyen and Vaikuntanathan (2016); Lutsko (2019); Fan and
Grunwald (2019); Carpenter and Grunwald (2020) but can exhibit key features of
real systems, including the formation of complex structures and kinetic traps
that impair assembly Thorkelsson  _et al._ (2015); Biancaniello  _et al._
(2005); Park  _et al._ (2008); Nykypanchuk  _et al._ (2008); Pfeifer and Saccà
(2018); De Yoreo  _et al._ (2015). The particular class of model we use here
has been shown to reproduce the thermodynamic and dynamic behavior of a range
of molecular and nanoscale assemblies at surfaces Whitelam  _et al._ (2014).
Such models provide a rigorous test of algorithmic control of self-assembly.

In order to allow thorough exploration of the self-assembly behavior
accessible to this class of model we express the interparticle potential and
time-dependent assembly protocol as arbitrary functions, encoded by neural
networks. In evolutionary language, which reflects the method of learning used
and provides a mnemonic for the role of each component of the algorithm, this
encoding is the instruction code or “genome” for self-assembling a material.
Molecular simulation carried out using the particle and protocols specified by
the genome results in the “phenome”, a material whose properties can be
measured and compared to a design goal.

Evolutionary learning on the parameters of the neural networks, called
neuroevolution Holland (1992); Fogel and Stayton (1994); Salimans  _et al._
(2017); Montana and Davis (1989); Such  _et al._ (2017); Whitelam  _et al._
(2020), can produce materials whose properties satisfy user-defined goals,
which can be directed or exploratory in nature. The algorithm needs no
information about possible candidate or competing structures, nor prior
knowledge of what constitutes a good self-assembly protocol or particle
design. For directed design we specify materials with certain pore geometries
and isolated clusters of certain sizes. The solutions identified by the
learning algorithm include interparticle interactions whose symmetries can be
realized by known molecules or complexes Swiegers and Malefetse (2002);
Bartels (2010); Elemans  _et al._ (2009). The approach is simple to implement
and can identify sophisticated design strategies, realizing complex structures
via hierarchical self-assembly pathways. It can also fail, if the design goal
is too challenging, and we discuss how to modify the goal in such cases.

Figure 1:  (a) We express the angular interaction potential and the time-
dependent self-assembly protocol for a set of model molecules in the form of
two neural networks, which together comprise the “genome” for making a
material. In images, attractive portions of particles are shown green. (b)
Upon specifying a design goal, neuroevolutionary learning can produce a genome
whose “phenome” – the result of molecular simulations carried out using the
potential and protocol specified by the genome – is a material satisfying that
goal.

Our approach is similar to that of Refs. Lindquist  _et al._ (2016); Jadrich
_et al._ (2017) in that we use an iterative learning method to promote self-
assembly, but differs in that we do not specify or build the target structure
in advance. In that respect it is similar to the approach of Ref. Miskin  _et
al._ (2016), and complementary to that work in that we express the design
problem differently (in the form of neural networks) and optimize differently
(via evolutionary methods). Our approach differs from other approaches to
inverse design in that we consider the design of particles and protocols
simultaneously, and do not appeal to physical concepts such as thermal
equilibrium or mechanical stability. We also go beyond traditional forms of
inverse design and borrow from the machine-learning literature to specify the
design goal of novelty Conti  _et al._ (2018). Within a space specified by
certain order parameters we instruct the evolutionary learning algorithm to
produce materials not seen previously, rather than materials with particular
properties. If we specify materials with 3- and 4-membered pores then novelty
search identifies structures dual to regular and semi-regular tilings of the
plane and motifs that comprise 2D quasicrystals. In this mode the algorithm
explores the space of what can be made rather than the space of structures
that are low in energy but not necessarily kinetically accessible.

Model and learning algorithm— We consider a class of coarse-grained model able
to capture the essential physics of molecular and nanoscale self-assembly at
surfaces Whitelam  _et al._ (2014). It comprises circular particles of hard-
core diameter $a$ on a two-dimensional square substrate of side $50a$. The
substrate has periodic boundary conditions in both directions. Particles
evolve under a stochastic dynamical protocol consisting of a grand-canonical
Monte Carlo algorithm with chemical potential $\mu$, which allows particles to
exchange with a notional solution Frenkel and Smit (1996), and the virtual-
move Monte Carlo algorithm Whitelam  _et al._ (2009); Hedges , which allows
particles to move on the surface according to an approximation of Brownian
motion Haxton  _et al._ (2015). Grand-canonical moves are proposed with
probability $1/(1+P)$, where $P$ is the instantaneous number of particles on
the surface Whitelam  _et al._ (2014). All trajectories start from distinct
disordered configurations consisting of 500 particles randomly deposited on
the surface, and are run for $t_{0}=10^{9}$ Monte Carlo steps.

The angular component of the interparticle attraction $U_{\bm{x}}(\theta)$ and
the time-dependent protocol $(\mu_{\bm{y}}(t),\epsilon_{\bm{y}}(t))$ are
encoded as neural networks, specified in Section S1. Simulation potentials
with angular dependence are often called “patchy” Zhang and Glotzer (2004).
The interaction potential reflects the idea that particles interact in a
complementary way, such as through DNA hybridization, hydrogen bonding, or
other directional donor-acceptor mechanisms Swiegers and Malefetse (2002);
Bartels (2010); Elemans  _et al._ (2009); Pfeifer and Saccà (2016). The
control parameters $\mu$ and $\epsilon$ influence the substrate density and
the strength of interparticle attractions, which could be done in experiment
by e.g. varying deposition rate and temperature (we call the process of
increasing $\epsilon$ “cooling”). We denote the parameters of the neural
networks by ${\bm{x}}$ and ${\bm{y}}$. Using evolutionary language we call
this encoding the material’s genome, an idea sketched in Fig. 1. The results
of molecular simulation, using the potential and protocol defined by the
genome, is the phenome.

To evolve genomes whose phenomes possess a desired property we use an
iterative genetic algorithm consisting of a population dynamics combined with
neuroevolution Holland (1992); Fogel and Stayton (1994); Salimans  _et al._
(2017); Montana and Davis (1989); Such  _et al._ (2017); Whitelam and Tamblyn
(2020), specified in Section S2. Neuroevolution, stochastic mutation of the
neural-network parameters, is equivalent in the limit of small mutations to
gradient descent in the presence of Gaussian white noise Whitelam  _et al._
(2020). The learning algorithm starts in Generation 0 with a population of 100
randomly-chosen genomes, and “expresses” their phenomes via $t_{0}$ steps of
the molecular simulation protocol described above. The algorithm identifies
the 10 phenomes possessing the largest values of an objective function $\phi$.
The 10 corresponding genomes are cloned and mutated in order to produce a new
population (or generation) of 100 genomes, whose phenomes are then expressed
via molecular simulation. This iterative procedure continues, generation by
generation, until terminated by the user.

The objective function or evolutionary pressure is a user-specified order
parameter $\phi$, evaluated at the final time point of each simulation. In
this work we consider order parameters built from two quantities. One is
$C_{k}$, the number of clusters of interacting particles of size $k$ (called
$k$-mers). The other is $N_{k}$, the number of convex loops of size $k$
(called $k$-gons) that can be drawn by joining the centers of interacting
particles. $k$-gons are pores: designing materials with specified pore sizes
is useful for e.g. gas separation Liu  _et al._ (2016).

Figure 2:  Evolutionary learning directed to maximize the number of 12-gons,
convex pores of 12 sides (shown light blue in images). The main panel shows
the yield $\phi$, as a function of generation $n$, produced by the most
successful genome (potential and protocol). The particle symbols are the data
points, which also show the form of the learned potential. Below the data
points are plots of the learned protocol (red and blue lines) and resulting
yield (green lines) as a function of time $t$. The format of those plots is
shown bottom right. Large positive values of $\epsilon$ indicate strong
particle attractions, and large positive and negative values of $\mu$ promote
dense and sparse substrates, respectively. The snapshots at the top show
portions of a simulation box from three different generations, indicated by
the blue lines. Parameters: $\phi_{0}=90$ 12-gons,
$\epsilon_{0}=\mu_{0}=20\,k_{\rm B}T$, $n_{0}=27$ generations, $t_{0}=10^{9}$
Monte Carlo steps. Figure 3:  The results of evolutionary learning (genome
bottom and phenome top) instructed to produce (a) 4-mers, (b) 5-mers and
5-gons, (c) 6-gons, and (d) 12-gons.

Directed search— We direct the algorithm to evolve a material containing
convex pores of size 12, and set $\phi=N_{12}$. This case provides an example
of an objective that is too complex to achieve without additional guidance:
12-gons are sufficiently complex that they do not form spontaneously under the
random particle- and protocol design that comprises the initial stage
(Generation 0) of the learning algorithm. All phenomes score zero, and the
learning algorithm has nothing to work with. In this case a simple
modification of the objective is sufficient to overcome the problem. We set
$\phi=N_{\min(x,12)}$, where $x$ is the size of the largest convex pore seen
across all 100 phenomes of a given generation. Thus if $x$ is 12 or larger
then the learning algorithm selects genomes that produce 12-gons; if $x$ is
smaller than 12 then it selects genomes that produce $x$-gons.

The results of several generations of learning using this objective are shown
in Fig. 2. The largest pore sizes seen in the first 4 generations are 10, 9,
11, and 11, and thereafter the first 12-gons are produced. The learning
algorithm improves its design and the yield of 12-gons over evolutionary time,
and eventually achieves the self-assembly of a structure dual to the 3.12.12
Archimedean tiling, which has one 3-gon and two 12-gons around each vertex
Grunbaum and Shephard (1977); Antlanger  _et al._ (2011); Whitelam (2016). To
do so requires a sophisticated design. The particle must present sticky
patches whose bisectors are separated (approximately) by angles $\pi/3$ and
$5\pi/6$. In addition, the patches must be inequivalent: if all patches
possess equal binding energy then kinetic traps impair the formation of the
structure Whitelam (2016). The solution identified by the learning algorithm
is to make one patch weaker than the other two, and to steadily cool the
substrate. The result is a hierarchical dynamics that starts with many
isolated 3-gons forming from the engagement of the strong patches. Eventually
the weak patches engage and cause the 3-gons to form a network, which
subsequently forms 12-gons [Fig. S2]. The learning algorithm also evacuates
the substrate, removing steric impediments to closure of the network. The
resulting strategy produces a yield of 12-gons superior to that achieved by a
human-designed particle and protocol [Fig. S3(a)].

In Fig. 3 we show the results of evolutionary learning instructed to make
4-mers, 5-mers and 5-gons, and 6-gons. In each case the strategy learned is
efficient: to produce 4-mers the algorithm evolves particles with two patches
separated by an angle $\pi/2$, leading to compact square clusters; to make
pentagonal 5-gons it evolves particles with patches separated by an angle
$3\pi/5$; and to make 6-gons the algorithm evolves particles with approximate
3-fold rotational symmetry, which self-assemble under the learned protocol
into the honeycomb lattice. Similar motifs are seen in a range of real system
that realize the honeycomb lattice Swiegers and Malefetse (2002); Bartels
(2010); Elemans  _et al._ (2009); Whitelam  _et al._ (2014). The evolutionary
pressure to achieve geometrical perfection of the interaction is relatively
weak: self-assembly of a particle with perfect three-fold rotational symmetry
and a learned time-dependent protocol results in a comparable yield of 6-gons
[Fig. S3(b)]. The evolutionary trajectories showing the emergence of these
designs are shown in Figs. S4, S5, and S6.

Figure 4:  The results of evolutionary learning instructed to produce novelty
in the space of 3-gons and 4-gons. The scatterplot shows the coverage obtained
in this space after 10 generations of learning; the images show examples of
the structures produced (the particles underlying the polygons are not shown).

Exploratory search— We end by showing that evolutionary learning can be used
in an exploratory mode, searching for novelty rather than to achieve a desired
property Conti  _et al._ (2018). To search for novelty within the space of
4-gons and 3-gons we impose the objective function

$\phi=\sum_{j}\sqrt{\frac{1}{4}\left(N_{3}-N_{3}^{(j)}\right)^{2}+\left(N_{4}-N_{4}^{(j)}\right)^{2}},$
(1)

where $j$ runs over all phenomes produced in all previous and current
generations. Maximizing (1) leads to an evolutionary pressure favoring
materials most unlike those produced to date, rather than materials with
specified values of $N_{3}$ and $N_{4}$. Over the course of 10 generations of
novelty search the learning algorithm produces the coverage of $(N_{3},N_{4})$
space shown in Fig. 4. Some of the polygon structures dual to the particle
structures found during that exploration are shown in the figure. These
include size-limited motifs; the square and triangle regular tilings of the
plane; the 3.6.3.6 and 3.3.3.4.4 Archimedean tilings Grunbaum and Shephard
(1977); Antlanger  _et al._ (2011); Whitelam (2016); and the $\sigma$, H, and
Z binding motifs prominent in dodecagonal quasicrystals van der Linden  _et
al._ (2012). The particle that gives rise to those motifs is shown in the
figure: it has irregular six-fold symmetry, different to the five-fold and
seven-fold coordination known to form similar motifs van der Linden  _et al._
(2012). A larger section of the material made by this particle is shown in
Fig. S7.

Conclusions— We have shown that a neuroevolutionary learning algorithm can
identify particles and protocols for the self-assembly of materials with
desired properties, without input from physical principles and with no prior
knowledge of self-assembly. The learning algorithm is capable of both directed
and exploratory design. It can assemble a material with a user-defined
property, or search for novelty in the space of specified order parameters. In
the latter mode it explores the space of what can be made rather than the
space of structures that are low in energy but not necessarily kinetically
accessible. Moreover, the approach described here can be used to address
design problems of considerable complexity: the neural-network encoding of
potential and protocol extends to an arbitrary number of inputs and outputs,
and evolutionary learning works with large numbers of parameters Such  _et
al._ (2017); Salimans  _et al._ (2017).

Acknowledgments – This work was performed as part of a user project at the
Molecular Foundry, Lawrence Berkeley National Laboratory, supported by the
Office of Science, Office of Basic Energy Sciences, of the U.S. Department of
Energy under Contract No. DE-AC02–05CH11231. I.T. acknowledges NSERC and
performed work at the NRC under the auspices of the AI4D and MCF Programs.

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##  S1 Neural-network encoding of potential and protocol

Interparticle attraction and time-dependent assembly protocols are encoded as
single-layer neural networks of $K=10^{3}$ hidden nodes, sketched in Fig. 1.
Particles possess an attractive interaction of range $a/10$. The angular
component of the interaction is the radial-basis-function neural network

$U_{\bm{x}}(\theta)=\sum_{\alpha=0}^{K-1}x_{3\alpha+1}\exp\left[-\frac{1}{2}\left(\frac{\theta-
x_{3\alpha+3}}{{\rm e}^{x_{3\alpha+2}}}\right)^{2}\right],$ (S1)

containing $N=3K$ trainable parameters $\bm{x}=\\{x_{1},\dots,x_{N}\\}$. If
$U_{\bm{x}}(\theta)$ lies outside the interval $[0,1]$ then it is set to the
appropriate edge of the interval. In images, angular portions of the particle
corresponding to $U_{\bm{x}}(\theta)>0$ are colored green, and are otherwise
blue. Initially (in Generation 0) we set $\exp(x_{3\alpha+2})=0.2$,
$x_{3\alpha+3}=2\pi\alpha/K$, and choose the
$x_{3\alpha+1}\sim{\mathcal{N}}(0,0.02)$ to be Gaussian random numbers. The
potential remains fixed for all time within each simulation. For computational
efficiency we evaluate the neural network only once, at the start of the
simulation, in order to create a lookup table for $U_{\bm{x}}(\theta)$ with
$\theta$ discretized over 1000 points on the interval $[0,2\pi)$.

Two particles $i$ and $j$ whose centers are a distance $d$ apart, where
$a<d\leq 11a/10$, experience an energy of interaction

$E_{ij}=-\epsilon\min(U_{\bm{x}}(\theta_{ij}),U_{\bm{x}}(\theta_{ji})).$ (S2)

Here $\epsilon>0$ sets the scale of the interaction. $\theta_{ij}\in[0,2\pi)$
is the angle (in an anti-clockwise sense) between two lines, the line joining
the center of particle $i$ to the point specified by $\theta=0$ on its
circumference, and the line joining the center of particle $i$ to the center
of particle $j$: see Fig. S1. The minimum function encodes the idea that
particles interact in a complementary way, such as through DNA hybridization,
hydrogen bonding, or other directional donor-acceptor mechanisms.

The time-dependent protocol $(\mu_{\bm{y}}(t),\epsilon_{\bm{y}}(t))$ is
encoded by a second neural network. Each trajectory starts with control-
parameter values $\epsilon=3\,k_{\rm B}T$ and $\mu=2\,k_{\rm B}T$. 1000 times
within each trajectory, at time increments of $10^{-3}t_{0}$, the control
parameters are set to the new values
$\epsilon\to\epsilon+\Delta\epsilon_{\bm{y}}(t)$ and
$\mu\to\mu+\Delta\mu_{\bm{y}}(t)$, where the neural network

$\displaystyle(\Delta\epsilon_{\bm{y}}(t),\Delta\mu_{\bm{y}}(t))$
$\displaystyle=$ $\displaystyle\frac{k_{\rm
B}T}{K}\sum_{\alpha=0}^{K-1}(y_{4\alpha+1},y_{4\alpha+2})$ (S3)
$\displaystyle\times$ $\displaystyle\tanh(y_{4\alpha+3}t+y_{4\alpha+4})$

contains $M=4K$ trainable parameters $\bm{y}=\\{y_{1},\dots,y_{N}\\}$. If
$\mu$ moves outwith the interval $[-20,20]$ it is returned to the appropriate
edge of the interval. Initially (in Generation 0) all parameters of this
network are Gaussian random numbers, $y_{i}\sim{\mathcal{N}}(0,1)$.

##  S2 Evolutionary learning algorithm

The evolutionary algorithm starts with 100 molecular simulations using
distinct genomes $(\bm{x},\bm{y})$, randomized as described in Section S1.
This set of 100 simulations is called Generation 0, and, following molecular
simulation, results in 100 phenomes. Let $\phi$ quantify the design goal (e.g.
the number of clusters or pores of a certain size). The algorithm selects the
10 genomes responsible for the phenomes having the 10 largest values of
$\phi$, choosing randomly in the event of equal scores. (For the novelty
search used to produce Fig. 4 we selected the top 25 phenomes, rather than the
top 10). To create the 100 genomes that comprise Generation 1 it draws 100
times randomly with replacement from this set of 10 genomes, and mutates each
by a set of Gaussian random numbers,

$\bm{x}\to\bm{x}+\delta\bm{x}\quad{\rm and}\quad\bm{y}\to\bm{y}+\delta\bm{y}.$
(S4)

Here $\delta\bm{x}=\\{\delta x_{1},\dots,\delta x_{N}\\}$ with $\delta
x_{i}\sim{\mathcal{N}}(0,\sigma_{x}^{2})$, and $\delta\bm{y}=\\{\delta
y_{1},\dots,\delta y_{M}\\}$ with $\delta
y_{i}\sim{\mathcal{N}}(0,\sigma_{y}^{2})$. The parameters $\sigma_{x}$ and
$\sigma_{y}$ are chosen independently for each simulation, as the absolute
value of the Gaussian random numbers ${\mathcal{N}}(0,0.1)$ and
${\mathcal{N}}(0,0.02)$, respectively. Doing so results in a combination of
many small mutations and the occasional large mutation. Molecular simulation
of this new set of 100 genomes results in the 100 phenomes of Generation 1,
and so on.

Figure S1:  Geometry for the interaction potential. The white arrow indicates
the line from the center of the particle to the point on its circumference
specified by $\theta=0$.

##  S3 Supplementary figures

Figure S2:  Self-assembly dynamics produced by the most successful genome
(potential and protocol, shown top left) from generation 27 of Fig. 2. $N_{3}$
and $N_{12}$ are the number of 3-gons (red) and 12-gons (light blue). The
snapshots are taken from the times indicated by the blue lines. 12-gons form
relatively late in the trajectory, emphasizing that self-assembly is a “sparse
reward” problem: the objective $\phi$ can be zero for most of the simulation.
Evolutionary methods that require information from only the final time point
of the simulation are natural ways to tackle such problems. Parameters:
$t_{0}=10^{9}$ Monte Carlo steps.

Figure S3: (a) Yield of the self-assembly process using the best generation-27
genome (particle and potential) of Fig. 2 (blue line) compared to the yield
obtained using a human-designed particle Whitelam (2016) combined with
evolutionary learning of the protocol (light blue). Also shown are three
simulations (red) using a human-designed particle and protocols, the latter
consisting of slow cooling at three rates at fixed chemical potential. (b)
Yield of the self-assembly process using the best generation-18 genome from
Fig. 3(c) and Fig. S6 (blue line), compared to that using a particle with
perfect three-fold rotational symmetry combined with evolutionary learning of
the protocol (light blue).

Figure S4:  Evolutionary learning with $\phi=C_{4}$, the number of 4-mers,
interacting clusters of 4 particles (shown darker in images). The format of
the figure is the same as Fig. 2. Parameters: $\phi_{0}=350$ 4-mers,
$\epsilon_{0}=\mu_{0}=20\,k_{\rm B}T$, $n_{0}=17$ generations, $t_{0}=10^{9}$
Monte Carlo steps.

Figure S5:  Evolutionary learning with $\phi=\min(C_{5},N_{5})$, the number of
5-mers (shown darker in images) and 5-gons (shown light blue). The format of
the figure is the same as Fig. 2, with the exception of the top-left image:
that shows the outcome of learning after 17 generations with $\phi=C_{5}$.
Evolution of pentagonal clusters requires the dual goal of 5-mers and 5-gons.
Parameters: $\phi_{0}=125$ 5-mers & 5-gons, $\epsilon_{0}=\mu_{0}=20\,k_{\rm
B}T$, $n_{0}=17$ generations, $t_{0}=10^{9}$ Monte Carlo steps.

Figure S6:  Evolutionary learning with $\phi=N_{6}$, the number of 6-gons,
convex pores of 6 interacting particles (shown light blue: see boxed image).
The format of the figure is the same as Fig. 2. Parameters: $\phi_{0}=700$
6-gons, $\epsilon_{0}=\mu_{0}=20\,k_{\rm B}T$, $n_{0}=17$ generations,
$t_{0}=10^{9}$ Monte Carlo steps.

Figure S7:  One of the materials identified by the novelty search of Fig. 4
shows the $\sigma$, H, and Z binding motifs characteristic of dodecagonal
quasicrystals van der Linden  _et al._ (2012). To produce this figure we used
the particle- and protocol design identified by the learning algorithm of Fig.
4, and ran a trajectory of length $20t_{0}$. The particles underlying the
pattern of 3-gons and 4-gons are not shown.