# Conjugacy classes of maximal cyclic subgroups and nilpotence class of
$p$-groups

M. Bianchi, R.D. Camina, & Mark L. Lewis

( )

###### Abstract

In this paper, we set $\eta(G)$ to be the number of conjugacy classes of
maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order
$p^{n}$ and nilpotence class $l$, then $\eta(G)$ is bounded below by a linear
function in $n/l$.
Keywords: group covering, nilpotence class, group exponent
2020 Mathematics Subject Classification: 20D15

##  1 Introduction

Unless otherwise stated, all groups in this paper are finite. Motivated by the
research area of classifying spaces of families of subgroups of infinite
groups, von Puttkamer asks [5, Question 5.0.9]: does the number of conjugacy
classes of maximal cyclic subgroups of a finite $p$-group for a prime $p>2$
grow with the order of the group? Clearly, the question is considering
noncyclic groups. More precisely, one can ask whether the number of conjugacy
classes of maximal cyclic subgroups of a noncyclic $p$-group of order $p^{n}$
is at least $n$ when $p$ is an odd prime. This work was initiated by X. Wu
asking the second author this question in a private communication.

The question can be phrased in terms of the size of a particular type of
covering of a group. Recall, a covering  of a group $G$ is a set of proper
subgroups $\\{H_{i}\\}$, called components, such that
$G\subseteq\bigcup_{i}H_{i}$. A covering is called irredundant if removing any
component means the set is no longer a covering. A normal covering of $G$ is a
covering that is invariant under conjugation by $G$. Note that the components
of a normal covering are not normal, but refers to the fact that the
components form orbits under conjugacy. Often, one takes only one
representative of each conjugacy class of components. Our question above
considers normal coverings where all the components of the cover are cyclic.
It is not difficult to see that the set of maximal cyclic subgroups is the
only irredundant covering by cyclic subgroups.

With this in mind, we introduce the following definition. Let $G$ be a finite
group. Denote the number of conjugacy classes of maximal cyclic subgroups of
$G$ by $\eta(G)$. We note that the questions of von Puttkamer and Wu have
negative answers. In [2], the second and third authors along with Yiftach
Barnea and Mikhail Ershov find for every prime $p>3$ and every integer $n\geq
3$, groups of order $p^{n}$ that all have $\eta=p+2$. For $p=2$, we also find
that $\eta(G)=3$ when $G$ is a dihedral $2$-group, a generalized quaternion
$2$-group, or a semi-dihedral $2$-group. For $p=3$, there is an infinite
family of $3$-groups with $\eta=9$. See Theorem 7.12 of [2].

On the other hand, we believe it is extremely rare for this to occur. In
particular, we prove that if we fix the nilpotence class of a $p$-group $G$,
then $\eta(G)$ will grow proportionally to $\log_{p}(|G|)$. The theorem we
prove is the following.

###### Theorem 1.1

Let $G$ be a noncyclic $p$-group of nilpotence class $l\geq 1$ and order
$p^{n}$. Then $\eta(G)\geq(p-1)(n/l-2)+p+1$.

The authors would like to thank Emanuele Pacifici for a number of helpful
conversations while working on this paper. The first author is partially
supported by INDAM-GNSAGA. We would like to thank Avinoam Mann for pointing
out [3].

##  2 Abelian groups

As noted in the introduction, when considering abelian groups calculating
$\eta(G)$ is the same as calculating the maximum number of subgroups in an
irredundant cyclic covering of $G$. In [4, Proposition 6 (ii)] the author
gives a formula, which involves the Euler totient function, for calculating
this number.

We take a slightly different approach; so we are including our results. We
will show for abelian $p$-groups that a function in $\eta(G)$ gives a lower
bound for $|G|$. We begin by computing $\eta(G)$ when $G$ is the direct
product of two cyclic $p$-groups.

We consider the set $G^{\\{p\\}}=\\{g^{p}\mid g\in G\\}$. For an element $g\in
G$, it is not difficult to see that that $\langle g\rangle$ is a maximal
cyclic subgroup of $G$ if and only if $g\in G\setminus G^{\\{p\\}}$.

###### Lemma 2.1

Suppose $G\cong C_{p^{a}}\times C_{p^{b}}$ with $a\geq b$. Then

$\eta(G)=p^{(b-1)}((a-b)(p-1)+p+1)\geq a+b.$

Proof. Let $G=\langle x\rangle\times\langle y\rangle$ with $x$ of order
$p^{a}$ and $y$ of order $p^{b}$. Let $C$ be a cyclic subgroup of $G$. Either
$C$ is a subgroup of $\langle(1,y)\rangle$ or there exist integers $n$ and $c$
so that $C$ is generated by $(x^{p^{n}},y^{c})$ for $0\leq n<a$ and $0\leq
c<p^{b}$. We claim that $C$ is maximal exactly when $n=0$, or $c$ is
relatively prime to $p$, or $C=\langle(1,y)\rangle$. To see this, observe that
in each of these cases, a generator of $C$ does not lie in $G^{\\{p\\}}$, so
$C$ is maximal. Furthermore, we claim that the generator for a maximal cyclic
subgroup is unique if we make the additional restriction that $0\leq
c<\min(p^{b},p^{a-n})$. First note for an integer $l$ that
$(x^{p^{n}},y^{c})^{1+lp^{a-n}}=(x^{p^{n}},y^{c+clp^{a-n}})$. Since $c$ is
coprime to $p$, as $l$ runs through the integers modulo $p^{b}$, then also
$cl$ will run through all of ${\mathbb{Z}}_{p^{b}}$. So we are getting as the
exponents for $y$, all elements in the coset $c+p^{a-n}{\mathbb{Z}}_{p^{b}}$
in ${\mathbb{Z}}_{p^{b}}$. It is not difficult to see that
$\\{0,1,...,p^{a-n}-1\\}$ is a transversal for $p^{a-n}{\mathbb{Z}}_{p^{b}}$
in ${\mathbb{Z}}_{p^{b}}$. Hence, there exists an integer $c^{\prime}$ with
$0\leq c^{\prime}\leq p^{a-n}-1$ so that
$c^{\prime}+p^{a-n}{\mathbb{Z}}_{p^{b}}=c+p^{a-n}{\mathbb{Z}}_{p^{b}}$. It
follows that $(x^{p^{n}},y^{c^{\prime}})$ will lie in
$\langle(x^{p^{n}},y^{c})\rangle$ and since it has the same order, it will be
a generator.

So, we claim the following are distinct maximal cyclic subgroups of $G$,

$\langle(x,y^{c})\rangle\;\;{\rm for}\;\;0\leq c\leq p^{b}-1\;\;{\rm
and}\;\;\langle(1,y)\rangle$ $\langle(x^{p^{n}},y^{c})\rangle\;\;\;{\rm
for}\;\;\;1\leq n<a,\;\;\;0\leq c<\min(p^{b},p^{a-n})\;\;\;{\rm and}\;\;{\rm
gcd}(c,p)=1.$

To count the number of subgroups of the form $\langle(x^{p^{n}},y^{c})\rangle$
we consider two cases, when $1\leq n\leq a-b$ and when $a-b+1\leq n<a$. For
$1\leq n\leq a-b$ we have $\phi(p^{b})=p^{b-1}(p-1)$ such subgroups where
$\phi$ is the Euler totient function. For $a-b+1\leq n\leq a-1$ we have
$\phi(p^{a-n})$ maximal cyclic subgroups of the form $\langle
x^{p^{n}},y^{c}\rangle$ where $p$ does not divide $c$ and $1\leq c\leq
p^{a-n}$. With this range on $b$, we obtain $1\leq a-n\leq b-1$. Observe that
$\phi(p^{a-n})=p^{a-n-1}(p-1)$. We can view this as $p^{i-1}(p-1)$ for $i$
running from 1 to $b-1$ or $p^{i}(p-1)$ for $i$ running from 0 to $b-2$. So,
in total we have

$p^{b}+(a-b)p^{(b-1)}(p-1)+\sum_{i=0}^{b-2}p^{i}(p-1)+1$
$=p^{b}+(a-b)p^{(b-1)}(p-1)+[(p^{(b-1)}-1)/(p-1)](p-1)+1$
$=p^{b}+(a-b)p^{(b-1)}(p-1)+p^{(b-1)}$ $=p^{(b-1)}((a-b)(p-1)+p+1)$

maximal cyclic subgroups, as desired. A simple inductive argument shows that
$p^{(b-1)}((a-b)(p-1)+p+1)\geq a+b$. $\Box$

It is useful to have a function which encodes this value. Let $p$ be a prime,
and let $a$ and $b$ be positive integers. We take $k={\rm max}(a,b)$ and
$l={\rm min}(a,b)$. We set $g_{p}(a,b)=p^{(l-1)}((k-l)(p-1)+p+1)$, so
$g_{p}(a,b)=\eta(G)$ when $G=C_{p^{a}}\times C_{p^{b}}$. In this next lemma,
we obtain the lower bound that we need to prove the general lower bound for
abelian groups.

###### Lemma 2.2

Let $a$ and $b$ be positive integers and $p$ a prime. Then
$g_{p}(a,b)\geq(p-1)(a+b-2)+p+1\geq a+b+1$.

Proof. Without loss of generality, we may assume $a\geq b$. Set $n=a+b$, so
that $a=n-b$. We work to show $g_{p}(a,b)\geq(p-1)(n-2)+p+1\geq n+1$. We have
$1\leq b\leq n/2$. We see that $g_{p}(n-b,b)=p^{b-1}((n-2b)(p-1)+p+1)$. We can
view $n$ as fixed, and this becomes a function in one variable:

$f(x)=g_{p}(n-x,x)=p^{x-1}((n-2x)(p-1)+p+1).$

We want to find the minimal value for $f(x)$ on $[1,n/2]$. We use calculus to
see that

$f^{\prime}(x)=p^{x-1}(((\ln p)(n-2x)-2)(p-1)+(\ln p)(p+1)).$

Setting this equal to $0$, we obtain the critical value of

$x=n/2+(p+1)/(2(p-1))-1/(\ln p).$

We claim that $(p+1)/(2(p-1))-1/(\ln p)$ is positive for all primes $p$. (This
can be shown using calculus or graphing using a computer.) Hence, this
critical point is not in our interval.

Now, $f(1)=((n-2)(p-1))+p+1=p(n-1)-n+3$ and $f(n/2)=p^{n/2-1}(p+1)$. Recall
that Fermat’s theorem tells us that any local extreme values would occur when
the derivative was zero. Since the derivative of this function is never zero
on this interval, we know that our minimum is the smallest of these two
values. Set $f_{1}(x)=((x-2)(p-1))+p+1$ and $f_{n/2}(x)=p^{x/2-1}(p+1)$ for
$x\geq 2$. Then $f_{1}(2)=f_{n/2}(2)$, but
$f^{\prime}_{n/2}(x)>f^{\prime}_{1}(x)$; so $f_{n/2}(x)>f_{1}(x)$ for $x>2$.
Thus, $g_{p}(n-b,b)\geq(p-1)(n-2)+p+1$. Finally, note that $(p-1)(n-2)+p+1\geq
n-2+3=n+1.$ $\Box$

We also need a refinement on the value of $\eta$ of a direct product of an
abelian $p$-group with a cyclic $p$-group.

###### Lemma 2.3

Suppose $G\cong H\times C_{p^{a}}$ where $H$ is an abelian $p$-group, then
$\eta(G)\geq(a+1)\eta(H)+1$.

Proof. We write $Y=\langle y\rangle\cong C_{p^{a}}$, so that $G=H\times Y$.
Let $\\{h_{i}\\}$, for $1\leq i\leq\eta(H)$, be representatives of generators
for the maximal cyclic subgroups of $H$. We now prove that
$\langle(h_{i},y^{p^{c}})\rangle$ for $0\leq c\leq a$ and $1\leq
i\leq\eta(H)$, along with $\langle(1,y)\rangle$ are distinct maximal cyclic
subgroups of $G$. This gives the required count for the lower bound.

First note that $h_{i}\in H\setminus H^{\\{p\\}}$, since $\langle
h_{i}\rangle$ is maximal in $H$, and thus, $(h_{i},1)$, $(h_{i},y^{p^{c}})$,
and $(1,y)$ all lie in $G\setminus G^{\\{p\\}}$ for $c\in\\{0,\dots,a\\}$. (It
is not difficult to see that $G^{\\{p\\}}=\\{(\alpha,\beta)\mid\alpha\in
H^{\\{p\\}},\beta\in\langle y^{p}\rangle\\}$.) Thus, each of these elements
generate a maximal cyclic subgroup of $G$. We need to show that they generate
different cyclic subgroups of $G$. Suppose
$\langle(h_{i},y^{p^{c}})\rangle=\langle(h_{j},y^{p^{d}})\rangle$ for $0\leq
c,d\leq a$. In both cases, by projecting into $H$, it is not difficult to see
that $\langle h_{i}\rangle=\langle h_{j}\rangle$, and thus, $i=j$. We are left
to consider $\langle(h_{i},y^{p^{c}})\rangle=\langle(h_{i},y^{p^{d}})\rangle$.
Projecting into $Y$, this forces $\langle y^{p^{c}}\rangle=\langle
y^{p^{d}}\rangle$, and thus, $c=d$ as required. $\Box$

We now obtain our lower bound on $\log_{p}(|G|)$ when $G$ is an abelian
$p$-group in terms of $\eta(G)$.

###### Theorem 2.4

Let $G$ be an abelian, noncyclic group of order $p^{n}$. Then

$\eta(G)\geq(p-1)(n-2)+p+1\geq n+1.$

Proof. We work by induction on $|G|$. Suppose first that $G=C_{p^{a}}\times
C_{p^{b}}$ for positive integers $a$ and $b$. By Lemma 2.1, we have
$\eta(G)=g_{p}(a,b)$ and applying Lemma 2.2, we have
$g_{p}(a,b)\geq(p-1)(a+b-2)+p+1\geq a+b+1$.

Now suppose $H$ is not cyclic and $|H|=p^{h}$ and $G=H\times C_{p^{a}}.$ We
have by induction, $\eta(H)\geq(p-1)(h-2)+p+1$.

By Lemma 2.3,

$\displaystyle\eta(G)=\eta(H\times C_{p^{a}})$ $\displaystyle\geq$
$\displaystyle(a+1)\eta(H)+1$ $\displaystyle\geq$
$\displaystyle(a+1)((h-2)(p-1)+p+1)+1$ $\displaystyle=$
$\displaystyle(a+1)(h-2)(p-1)+a(p+1)+p+1+1$ $\displaystyle=$
$\displaystyle(a+1)(h-2)(p-1)+a(p-1)+2a+p+2$ $\displaystyle=$
$\displaystyle((a+1)(h-2)+a)(p-1)+2a+p+2$ $\displaystyle\geq$
$\displaystyle(h-2+a)(p-1)+p+1$

since $n=h+a$, this gives the result.

Finally, note that $(p-1)(n-2)+p+1\geq n-2+3=n+1.$ $\Box$

When $p$ is odd, we can improve the inequality in Theorem 2.4.

###### Corollary 2.5

If $p$ odd and $G$ is a noncyclic, abelian group of order $p^{n}$, then
$\displaystyle\eta(G)\geq(\frac{p+1}{2})n$.

Proof: We apply the result from Theorem 2.4:

$\displaystyle(p-1)(n-2)+p+1$ $\displaystyle=$ $\displaystyle(p-1)n-2p+2+p+1$
$\displaystyle=$ $\displaystyle(p+1)n/2+(p-3)n/2-p+3$ $\displaystyle=$
$\displaystyle(p+1)n/2+(p-3)(n-2)/2.$

Since $p\geq 3$ and $n\geq 2$, this is at least $(p+1)n/2$. $\Box$

##  3 $\eta$ and nilpotence class

We now show that if we fix the nilpotence class of a $p$-group $G$, then a
function of $\eta(G)$ gives a lower bound for $|G|$. We will see that this can
be viewed as generalization of Theorem 2.4 for abelian groups. We define the
terms of the lower central series of $G$ inductively as follows, $G^{1}=G$ and
for $i\geq 1$, we set $G^{i+1}=[G^{i},G]$. Observe that $G$ is nilpotent of
nilpotence class $l$ if $G^{l+1}=1$ and $G^{l}>1$.

We first prove a lemma regarding the order of elements of $G^{l}$ when $G$ has
nilpotence class $l$. Taking $|G|=p^{n}$, note that $n\geq l+1$, so $\lfloor
n/(l+1)\rfloor\geq 1$. It has been suggested that this next lemma is related
to [3].

###### Lemma 3.1

Let $G$ be a $p$-group of nilpotence class $l\geq 2$ and order $p^{n}$. Then
$G^{l}$ has exponent dividing $p^{\lfloor n/(l+1)\rfloor}$.

Proof. We work by induction on $l$. We begin with the case that $l=2$. If
$|G^{2}|\leq p^{n/3}$, then we have the result. Thus, we may assume that
$|G^{2}|>p^{n/3}$ and so, $|G/G^{2}|<p^{2n/3}$. We can find
$a_{1},\dots,a_{k}\in G$ so that $G/G^{2}=\langle
a_{1}G^{2}\rangle\times\langle a_{2}G^{2}\rangle\times\cdots\times\langle
a_{k}G^{2}\rangle$. We know that $G^{2}$ is central, and so it is abelian.
Also, it is generated by $\\{[a_{i},a_{j}]\mid 1\leq i<j\leq k\\}$. Since $G$
is a $p$-group, it suffices to show that $o([a_{i},a_{j}])$ is less than or
equal to $p^{n/3}$ for all $1\leq i<j\leq k$. Observe that
$o(a_{i}G^{2})o(a_{j}G^{2})$ divides $|G/G^{2}|<p^{2n/3}$, and so, without
loss of generality $o(a_{i}G^{2})$ is less than or equal to $p^{n/3}$. Now,
$[a_{i},a_{j}]^{\lfloor n/3\rfloor}=[a_{i}^{\lfloor n/3\rfloor},a_{j}]=1$.
This completes the proof when $l=2$.

We now assume that $l\geq 2$. If $|G^{l}|\leq p^{n/(l+1)}$, then the result
holds. Thus, we may assume that $|G^{l}|>p^{n/(l+1)}$, and so,
$|G/G^{l}|<p^{nl/(l+1)}$. We see that $G^{l}$ is generated by the set
$\\{[a,b]\mid a\in G^{l-1},b\in G\\}$. As above, since $G^{l}$ is central
(thus abelian) and a $p$-group, it suffices to show that $o([a,b])\leq
p^{n/{(l+1)}}$ when $a\in G^{l-1}$ and $b\in G$. By induction, we know since
$G/G^{l}$ has nilpotence class $l-1$ that $o(aG^{l})$ is less than or equal to
$p^{(nl/(l+1))/l}=p^{n/(l+1)}$. We then have that $[a,b]^{\lfloor
n/(l+1)\rfloor}=[a^{\lfloor n/(l+1)\rfloor},b]=1$. This proves the desired
result. $\Box$

We now have what we need to prove the theorem. Notice that when $l=1$, $G$ is
abelian and the inequality is the inequality that was proved in Theorem 2.4.
Thus, we can use the abelian case as the base case for our induction. We do
need two results that we have proved in [1]: (1) If $N$ is a normal subgroup
of $G$, then $\eta(G)\geq\eta(G/N)$. (2) If $G$ is a group, then
$\eta(G)\geq\eta(Z(G))$.

Proof. [Proof of Theorem 1.1] We work by induction on $l$. If $l=1$, then this
is Theorem 2.4.

We now suppose that $l\geq 2$. Note that $G/G^{\prime}=G/G^{2}$ is not cyclic,
so the induction hypothesis is valid even if $l=2$. If $|G/G^{l}|\geq
p^{n(l-1)/l}$, then by the induction hypothesis we have

$\eta(\frac{G}{G^{l}})\geq(p-1)\left(\frac{n(l-1)/l}{l-1}-2\right)+p+1=(p-1)\left(\frac{n}{l}-2\right)+p+1.$

As we noted before the start of the proof, we have that
$\eta(G)\geq\eta(G/G^{l})$, and so we have the conclusion in this case.

Thus, we may assume that $|G/G^{l}|<p^{n(l-1)/l}$, and so,
$|G^{l}|>p^{(n/l)}>p^{n/(l+1)}$. By Lemma 3.1, this implies that $G^{l}$ is
not cyclic. It follows that $|Z(G)|>p^{n/l}$ and is not cyclic. By Theorem
2.4, we have $\eta(Z(G))\geq(p-1)(n/l-2)+p+1$, and as we noted before the
start of the proof, we have $\eta(G)\geq\eta(Z(G))$. This yields the result
when $n=l$ and proves the theorem. $\Box$

Rewriting the inequality in Theorem 1.1, we obtain
$\ln_{p}(|G|)\leq((\eta(G)-p-1)/(p-1)+2)l$; so Theorem 1.1 can be viewed as
saying that that the size of $G$ is bounded in terms of a function of
$\eta(G)$ and the nilpotence class of $G$. I.e., if the order of $G$ is
growing, then either $\eta$ or $l$ must be growing. If one fixes $|G|$, then
the nilpotence class being large forces $\eta$ to be small and vice versa. In
this way, it is like the uncertainty principle.

## References

  * [1] M. Bianchi, R. D. Camina, Mark L. Lewis, E. Pacifici, Conjugacy classes of maximal cyclic subgroups, Preprint.
  * [2] Yiftach Barnea, Rachel D. Camina, Mikhail Ershov, Mark L. Lewis, Preprint.
  * [3] T. E. Easterfield, The orders of products and commutators in prime-power groups, Proc. London Math. Soc. (2), 36 (1933), 73-77.
  * [4] J. R. Rogério, A note on maximal coverings of groups, Comm. Algebra 42(10) (2014), 4498-4508.
  * [5] Timm Wilhelm von Puttkamer, On the Finiteness of the Classifying Space for Virtually Cyclic Subgroups, PhD thesis.

Mariagrazia Bianchi: Dipartimento di Matematica F. Enriques, Università degli
Studi di Milano, via Saldini 50, 20133 Milano, Italy.
<EMAIL_ADDRESS>
Rachel D. Camina: Fitzwilliam College, Cambridge, CB3 0DG, UK.
<EMAIL_ADDRESS>
Mark L. Lewis: Department of Mathematical Sciences, Kent State University,
Kent, Ohio, 44242 USA.
<EMAIL_ADDRESS>


#  Fin-Fact: A Benchmark Dataset for Multimodal Financial Fact Checking and
Explanation Generation

Aman Rangapur Haoran Wang Kai Shu
Illinois Institute of Technology, Chicago, IL, USA
{arangapur<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fact-checking in financial domain is under explored, and there is a shortage
of quality dataset in this domain. In this paper, we propose Fin-Fact, a
benchmark dataset for multimodal fact-checking within the financial domain.
Notably, it includes professional fact-checker annotations and justifications,
providing expertise and credibility. With its multimodal nature encompassing
both textual and visual content, Fin-Fact provides complementary information
sources to enhance factuality analysis. Its primary objective is combating
misinformation in finance, fostering transparency, and building trust in
financial reporting and news dissemination. By offering insightful
explanations, Fin-Fact empowers users, including domain experts and end-users,
to understand the reasoning behind fact-checking decisions, validating claim
credibility, and fostering trust in the fact-checking process. The Fin-Fact
dataset, along with our experimental codes is available at
https://github.com/IIT-DM/Fin-Fact/.

##  1 Introduction

In an era characterized by the rapid spread of misinformation and the
proliferation of fake news, fact-checking has emerged as a critical tool for
ensuring the accuracy and reliability of information Saakyan et al. (2021);
Wadden et al. (2020); Sarrouti et al. (2021). The emergence of social media
platforms and the wide accessibility of multimodal content have intensified
the complexities linked with verifying the accuracy of assertions Mishra et
al. (2022). Notably, the financial sector introduces its distinctive array of
difficulties, given that precise and timely data plays a pivotal role in
enabling well-informed investment choices and upholding market stability.
Additionally, financial fact-checking encounters specific hurdles, such as the
need for customized data to address unique requirements and nuances.
Furthermore, the manipulation of images to exploit visualization bias presents
another significant challenge in the verification process Mansoor and Harrison
(2018).

Figure 1: A demonstration of comprehensive multimodal fact-checking and the
creation of explanations.

The rise of misinformation in the financial domain has become a pressing
concern, with potential impacts on public trust, investor decisions, and
overall market stability Kogan et al. (2019); Clarke et al. (Forthcoming); Zhi
et al. (2021); Liu and Moss (2022). To counter the spread of misleading
information, fact-checking methods have gained importance in financial
reporting and analysis Zhi et al. (2021); Mohankumar et al. (2023). However,
the development of effective and dependable models in this domain has been
hindered by the lack of suitable benchmark datasets that accurately represent
the intricacies of financial information and context.

Table 1: An example from Fin-Fact

Claim |  Student debt relief is taxable by the Internal Revenue Service.
---|---
Author |  Jeff Cercone
Posted |  08/30/2022
Justification |  The up to $20,000 of federal student loan forgiveness announced by Joe Biden wont…
Evidence |  The post shows a screenshot of an IRS web page about canceled debt, with this sentence…
Image |  https://politifact.com/photos/2a603743e.jpg
Caption |  President Joe Biden speaks about student loan debt forgiveness…
Visual Bias |  0
Issues |  Debt, Taxes
Label |  True

In recent years, there has been notable progress in creating various datasets
for fact-checking Wadden et al. (2020); Wadden and Lo (2021); Wadden et al.
(2022); Sarrouti et al. (2021); Saakyan et al. (2021). However, there is a
noticeable gap in addressing the unique demands of fact-checking within the
financial domain. Financial fact-checking faces several significant
challenges. Firstly, it requires meticulously curated data that can encompass
the intricate nuances of financial discourse. Financial documents and
journalistic pieces often employ specialized language that differs from
conventional structures. However, existing datasets frequently lack
comprehensive coverage of financial news articles, and the absence of expert
annotations diminishes the reliability of the data. Secondly, financial data
is highly context-sensitive and constantly evolving, emphasizing the need for
a dataset that can accurately capture the dynamic nature of financial markets.
Lastly, the landscape of financial fact-checking introduces the challenge of
visualization bias, where deliberate manipulation of visual content can shape
perception and distort the accuracy of claims.

In this paper, we tackle the challenge of compiling, annotating, and refining
a comprehensive corpus of financial texts that faithfully represents financial
reporting, accounting methodologies, and market fluctuations. The realms of
financial fact-checking and explanation generation present distinct obstacles
that require specialized approaches. The necessity for tailored data capable
of navigating financial terminology and intricate visual elements underscores
the interdisciplinary nature inherent in this research endeavor. Figure 1
illustrate a comprehensive multimodal fact-checking and the creation of
explanations while Table 1 displays an example instance from the corpus.

Presenting Fin-Fact, a novel benchmark dataset specifically designed for
multimodal financial fact-checking and explanation generation. Our key
contributions and discoveries are as follows:

  * •

We introduce Fin-Fact, the inaugural benchmark dataset designed for verifying
claims within the financial domain. This dataset encompasses 3,562 claims,
each accompanied by expert fact-checkers’ justifications.

  * •

Fin-Fact holds the potential for explanation generation, facilitated by the
inclusion of authoritative ruling comments from skilled fact-checking
professionals.

  * •

Our investigation reveals performance challenges in applying the state-of-the-
art models to Fin-Fact in open-domain contexts, underlining the need for
improved generalization.

##  2 Related Work

Fact Checking. Significant efforts have been dedicated to creating fact-
checking datasets for automated fact-checking systems Wadden et al. (2020,
2022); Thorne et al. (2018); Saakyan et al. (2021). Previous studies have
predominantly focused on predicting the accuracy of claims from diverse
sources. While large-scale datasets from various domains have been utilized
Gupta and Srikumar (2021), they might not be suitable for identifying
misinformation related to financial matters due to domain-specific
disparities.

Although general-content misinformation datasets are readily accessible, only
a limited number of datasets pertain to online financial misinformation Clarke
et al. (Forthcoming); Hossain et al. (2020); Kogan et al. (2019); Zhi et al.
(2021); Liu and Moss (2022); Zhang et al. (2022); Boehm and Kroner (2023).
Current financial misinformation datasets lack clear labeling and
justifications, raising concerns about result reliability. In contrast, the
Fin-Fact dataset is distinct with genuine data and a multimodal structure,
combining text and images to encompass a wide range of financial information.
Additionally, it includes expert fact-checker comments, enabling comprehensive
explanations by models.
Explanation Generation. Explanation generation plays a pivotal role in
facilitating human comprehension of claim credibility. It involves leveraging
external knowledge graphs to create semantic traces originating from the claim
itself Gad-Elrab et al. (2019); Li et al. (2020); Sarrouti et al. (2021).
These semantic traces function as explanations that substantiate the veracity
of claims. This approach offers valuable insights into the rationale behind
the model’s decision-making, thereby fostering trust. Moreover, the process of
explanation generation relies on drawing evidence from diverse sources Thorne
et al. (2018); Hanselowski et al. (2019); Fan et al. (2020) to validate
claims. However, this evidence is frequently comprised of isolated sentences
extracted from extensive collections of documents, which can make it
challenging for humans to interpret the broader context. To effectively
generate explanations, a high-quality dataset annotated by humans is
essential. This paper addresses the need for such a dataset.

##  3 The Fin-Fact Dataset

The Fin-Fact dataset presents a diverse array of labels that enhance the depth
of analysis when evaluating financial claims. These labels contribute a
multifaceted perspective to the fact-checking process, augmenting its
analytical capabilities.

At the core of the dataset are the ‘Claim’ and ‘Author’ labels, which
respectively denote the primary assertion and its originating source. The
inclusion of the ‘Posted Date’ attribute introduces temporal context, while
the ‘Sci-digest’ label provides the summary of the claim. Further
contextualization is achieved through the ‘Justification’ label, elucidating
the accuracy of the claim, and the ‘Evidence’ label, which presents
corroborative information connected through ‘Evidence link.’ The dataset also
acknowledges the visual dimension through ‘Image link’ and ‘Image Caption.’
Critically, the ‘Visualisation Bias Label’ evaluates potential biases linked
to images. Complexities inherent in the claims are highlighted by the ‘Issues’
label, while the ultimate assessment is provided by the "Claim Label",
offering a definitive classification of "True", "False", or "NEI (Not Enough
Information)".

By amalgamating these labels, the dataset establishes a comprehensive and
multidimensional resource. This resource accommodates textual, temporal,
evidentiary, and visual components, all of which are imperative for a thorough
evaluation of claims during the fact-checking process.

###  3.1 Data Collection

PolitiFact111http://politifact.com/ and FactCheck222http://factcheck.org are
prominent online platforms dedicated to countering the spread of false
information. These platforms engage skilled fact-checkers to meticulously
analyze and verify individual claims, subsequently producing articles that
offer their conclusions supported by relevant evidence. In our study, we
leveraged these platforms as our primary sources of data. Specifically, we
devised a comprehensive process to extract essential information from these
platforms.

To elaborate, we devised a systematic process to gather essential information
from PolitiFact and FactCheck. This encompassed the extraction of text-based
claims and the assignment of corresponding truthfulness labels. Moreover, we
retrieved both textual and visual evidence, along with their associated links,
which contributed substantively to the assessment of claim accuracy.

It’s noteworthy that the initial claims were collected by journalists
affiliated with these platforms. These claims originated from diverse sources,
including online speeches, public statements, news articles, and social media
posts. Importantly, the fact-checkers from these platforms played a pivotal
role by providing truthfulness labels, pertinent evidence, references to
corroborating sources, and the articles delivering their final verdict. This
comprehensive approach ensured the thorough and reliable collection of data,
reinforcing the credibility of our assessment of claim accuracy.

###  3.2 Dataset Statistics

The Fin-Fact dataset is an encompassing compilation of claims within the
financial domain, spanning diverse sectors such as Income, Finance, Economy,
Budget, Taxes, and Debt, as visualized in Figure 2. This dataset is
meticulously crafted, comprising a total of 3,562 claims, curated to
encapsulate the intricacies inherent in financial discourse.

Figure 2: Diverse sectors within the Fin-Fact dataset.

In the Fin-Fact dataset, claims are categorized into three labels: "True",
"False", and "NEI (Not Enough Information)" representing the veracity of each
claim in the financial domain. The dataset contains 1,807 ‘True’ claims that
are verified as accurate, 1,315 ‘False’ claims that have been proven
inaccurate through fact-checking procedures, and 440 ‘NEI’ instances where
there is insufficient evidence to make a determination. With its comprehensive
span across a variety of claims, diverse sectors, and an equitable
distribution of labels, the Fin-Fact dataset stands as a robust cornerstone
for the development, assessment, and progression of fact-checking models in
the domain of finance.

##  4 Experimental Results

In this series of experiments, the primary focus revolved around evaluating
the accuracy of Natural Language Inference (NLI) models for fact-checking
tasks. The assessment encompassed a range of prominent models, including
ELECTRA Clark et al. (2020), BART Lewis et al. (2020), RoBERTa Liu et al.
(2019), and GPT-2 Radford et al. (2019). Each model underwent scrutiny using
the Fin-Fact dataset, enabling an assessment of their effectiveness in
distinguishing financial claims. The outcomes of these fact-checking endeavors
provided thought-provoking insights: ELECTRA demonstrated 29%, BART-Large
achieved 33%, RoBERTa-Large showcased 32%, and GPT-2 emerged as the leader
with an accuracy of 43% as shown in Table 2. These findings underscore the
intricate challenges posed by financial fact-checking, with models displaying
varying degrees of performance within this domain. Figure 3 illustrates the
graph that compares the performance of ANLI models on Fin-Fact.

Model | Precision | Recall | F1 Score | Accuracy
---|---|---|---|---
BART | 0.369 | 0.344 | 0.300 | 0.331
RoBERTa | 0.393 | 0.346 | 0.285 | 0.318
ELECTRA | 0.351 | 0.330 | 0.276 | 0.287
GPT-2 | 0.347 | 0.337 | 0.312 | 0.430

Table 2: Performance of Fin-fact on ANLI models.

The final phase of experimentation delved into the intricate realm of
generating explanations for the claims. For each claim in the dataset, we
employed the BART model to generate explanations, extracting insights that
highlight the key factors contributing to the determination of claim accuracy.
These explanations were obtained using the justifications provided to the
claim. To quantitatively evaluate the quality of these explanations, we
leveraged the GLUE and ROUGE metrics as shown in Table. 3. The Evidence label
in the dataset served as the ground truth, enabling us to assess the alignment
between the generated explanations and the human-provided justifications.

Figure 3: Comparison of Scores for ANLI Models.

Model | ROUGE-1 | ROUGE-2 | ROUGE-3 | GLUE
---|---|---|---|---
BART | 0.84 | 0.63 | 0.46 | 0.062

Table 3: Performance scores of Fin-fact on BART model for explaination
generation.

While our primary focus was on evaluating NLI models for fact-checking and
explanation generation, the Fin-Fact dataset offers versatility for various
applications. Researchers in multimodal machine learning can leverage it as a
valuable benchmark. With its unique combination of textual and visual
financial data, Fin-Fact provides an ideal testbed for experimenting with
state-of-the-art multimodal models. Users are encouraged to assess and enhance
these models, contributing to advancements in this critical area.

##  5 Conclusion and Future Work

The emergence of Fin-Fact signifies a pivotal advancement in the quest to
counter misinformation within the financial domain. By providing expert
annotations, comprehensive claims, and the potential for explanatory insights,
Fin-Fact empowers fact-checking systems to achieve heightened levels of
precision and transparency. Its interdisciplinary design tackles financial
language intricacies and evolving contextual complexities. This construct
serves as a robust cornerstone for more effective and reliable fact-checking
processes.

In the realm of finance, Fin-Fact fosters trust, bolsters credibility, and
aids fact-checking. We’re committed to addressing visualization bias, a
prominent challenge, and exploring the impact of manipulated images on claim
interpretation. We plan to extract quantitative insights from media, enhancing
the dataset with tables and charts. Our goal is to release an improved
dataset, offering a more comprehensive resource for researchers, fact-
checkers, and decision-makers.

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##  Appendix A Additional Details of Dataset

Data visualization plays a pivotal role in the communication of information
and insights. However, in the context of misinformation, data visualization
can become a potent tool for distorting facts and promoting false narratives.
The effectiveness and accuracy of visualizations can be compromised when the
images are manipulated.

To address the challenges posed by visualization bias and its potential impact
on the interpretation of claims, we have incorporated an additional label,
‘Visualisation Bias Label’ into the Fin-Fact dataset. This label was manually
human-annotated to evaluate the presence of any visualization bias in the
images associated with claims or captions of the images. Our commitment to
transparency and comprehensive analysis led us to include this label, enabling
researchers to examine instances of potential manipulation and its influence
on visual content.

The ‘Visualisation Bias Label’ serves as a vital component of our dataset,
contributing to a more comprehensive understanding of the interaction between
textual claims and accompanying images. By providing insights into the
potential manipulation of visual content, we contribute to the efforts aimed
at fostering accurate fact-checking and informed decision-making in the realm
of financial information.

##  Appendix B Additional Evaluation Details

In addition to the primary evaluation metrics discussed earlier, we believe in
providing a comprehensive assessment of our models’ capabilities. To achieve
this, we delve deeper into the performance analysis, offering a more nuanced
understanding of our models’ effectiveness. This comprehensive evaluation
includes the utilization of confusion matrices and classification reports,
specifically tailored for the Natural Language Inference (NLI) models that
play a pivotal role in our fact-checking endeavors. We employ confusion
matrices to scrutinize our NLI models’ performance at a finer level of detail.
Figure 4 shows the confusion matrices of the state-of-the-art ANLI models.

(a) Confusion matrix of BART on Fin-Fact.

(b) Confusion matrix of ELECTRA on Fin-Fact.

(c) Confusion matrix of RoBERTa on Fin-Fact.

(d) Confusion matrix of GPT-2 on Fin-Fact.

Figure 4: Confusion matrices of different ANLI models on Fin-Fact.

Beyond confusion matrices, we present comprehensive classification reports for
each of our NLI models. These reports go beyond mere accuracy metrics and
provide in-depth assessments of the model’s performance. For every label
(‘True,’ ‘False,’ and ‘NEI’), we furnish precision, F1-score, and recall
values. Such detailed reports not only offer a holistic view of each model’s
overall accuracy but also highlight its strengths and weaknesses in
classifying claims within the intricate domain of finance.

Table 4: Classification Metrics for Various Models

Model | Precision | Recall | F1-Score | Label
---|---|---|---|---
BART | 0.51 | 0.42 | 0.46 | True
| 0.12 | 0.44 | 0.19 | NEI
| 0.48 | 0.17 | 0.25 | False
RoBERTa | 0.50 | 0.42 | 0.46 | True
| 0.12 | 0.50 | 0.20 | NEI
| 0.55 | 0.12 | 0.20 | False
ELECTRA | 0.49 | 0.31 | 0.38 | True
| 0.12 | 0.50 | 0.19 | NEI
| 0.44 | 0.18 | 0.25 | False
GPT-2 | 0.49 | 0.71 | 0.58 | True
| 0.14 | 0.17 | 0.15 | NEI
| 0.41 | 0.13 | 0.20 | False

##  Appendix C Evaluation Setup and Reproducibility

###  C.1 Hyperparameter Settings

In the process of evaluating the corpus with Natural Language Inference (NLI)
models using the Transformers library from Hugging Face, we meticulously
explored various hyperparameter settings to ascertain the most effective
configuration. Among these settings, the max_length parameter of the tokenizer
was subjected to thorough experimentation. We experimented with different
values to strike a delicate balance between capturing crucial information and
ensuring efficient processing. After rigorous testing, it became evident that
a max_length value of 256 yielded the most promising results. This choice was
the culmination of a systematic exploration, aiming to align our settings with
industry best practices and to extract optimal performance from the models.

###  C.2 Computing Infrastructure

The conducted experiments were carried out leveraging the computational
prowess of the NVIDIA RTX 3070 GPU with 16 GB RAM, meticulously chosen for its
robust processing capabilities. The overarching objective of these experiments
was to meticulously evaluate the performance of a diverse spectrum of models
on the Fin-Fact dataset. This dataset was thoughtfully curated to encompass an
extensive array of financial claims spanning various sectors, including Trade,
Capital & Trade, Financial Regulations, Economy, State Budget, Taxes, and
Debt. The computational resources furnished by the NVIDIA RTX 3070 GPU served
as a cornerstone, ensuring the dependable and efficient execution of the
experimental procedures. The average evaluation time taken by NLI models was
45 minutes, while the explanation generation model took 4 hours.