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Sumana Hatui

Publications and source records attributed to Sumana Hatui.

17 recordsLinked to original sources
arXiv2026

On the Faithful Projective Representations of Finite Groups and their Minimal Dimension

The first part of this article is devoted to characterizing the cocycles $\alpha$ of a finite group $G$ that give rise to faithful projective representations of $G$. We prove that a $p$-group $G$ admits a faithful irreducible projective representation if and only if the cohomology class $[\alpha]$ does not lie in the image of the inflation map $\operatorname{inf}: \mathrm{H}^2\!\left(G / N, \mathbb{C}^{\times}\right) \longrightarrow \mathrm{H}^2\!\left(G, \mathbb{C}^{\times}\right)$ for any non-trivial central subgroup $N$ of $G$. In the case where $[\alpha] \in \operatorname{Im}(\operatorname{inf})$, we determine a criterion such that a direct sum of irreducible $\alpha$-representations is faithful. We conclude this part by describing the behaviour of cocycles $\alpha$ that yield faithful irreducible representations for direct products of groups. In the second part, we introduce the notion of the projective embedding degree of a finite group $G$, defined as the smallest integer $n$ such that $G$ embeds into $\mathrm{PGL}_n(\mathbb{C})$; equivalently, it is the smallest $n$ such that $G$ has a faithful complex projective representation of degree $n$. We also define the analogous notion of the irreducible projective embedding degree of $G$. These invariants have been investigated for several classes of groups, including direct products of groups, finite abelian groups, extra-special $p$-groups, Heisenberg groups, and groups of order $p^3$, $p^4$ (for primes $p$), and $p^5$ (for $p \geq 5$).

math.RT↗
arXiv2026

On the Schur multiplier of $p$-groups with abelianization $s$-elementary abelian

Let $p$ be an odd prime. We describe a method to compute the Schur multiplier of finite $p$-groups $G$ of nilpotency class $2$ such that $G/[G,G]$ is isomorphic to direct product of copies of $\mathbb{Z}_{p^s}$ for $s \in \mathbb{N}$, generalizing a method of Blackburn and Evens, who treated the case $s=1$. As an application, we investigate which abelian $p$-groups can occur as the Schur multiplier of a non-abelian $p$-group. We further introduce the notions of $s$-special $p$-groups of rank $k$ generalizing the notion of special $p$-groups of rank $k$. We study the structural properties, compute the Schur multipliers of $s$-special $p$-groups of rank $1$.

math.GR↗
arXiv2025

On the deep commuting graph of a finite group

Let $G$ be a finite group and let $\tilde{G}$ be a Schur cover of $G$. The deep commuting graph $\Delta_D(G)$ of $G$ is a simple graph with vertex set $G$, where two distinct vertices are adjacent if their pre-images commute in $\tilde{G}$. The deep commuting graph of a finite group was first introduced in [P. J. Cameron and B. Kuzma, Between the enhanced power graph and the commuting graph, {\it J. Graph Theory} {\bf 102} (2023), no. 2, 295--303], where the authors have shown that $\Delta_D(G)$ is fixed irrespective of the choice of the Schur cover $\tilde{G}$. In this paper, we first prove that $\Delta_D(G)$ is complete if and only if $G$ is cyclic. Also, we classify finite simple groups, symmetric groups and alternating groups, for which $\Delta_D(G)$ is perfect. In addition, explore several other properties of $\Delta_D(G)$ like Eulerianess, universality and connectedness of reduced deep commuting graphs. Next, we classify the finite abelian groups for which deep commuting graphs coincide with enhance power graphs. We also characterize the dominant vertices for the deep commuting graphs of finite abelian groups and examine the connectedness of the associated reduced deep commuting graphs. These properties of the deep commuting graphs for the non abelian groups like symmetric groups, alternating groups, dihedral groups, generalized quaternion group and Heisenberg groups are also discussed.

math.GR↗
arXiv2025

Finite groups with nearly half as many cyclic subgroups as elements

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American Mathematical Monthly 122.3 (2015): 275-276], an open problem was asked to classify the groups $G$ with $|C(G)|=|G|-r$, where $2 \leq r \leq |G|-1$. In this article, first we show that, for an odd prime $p$, there are infinitely many groups $G$ with $|C(G)|= \frac{|G|}{2}$, $|C(G)|=\frac{|G|}{p^{q-1}}$ (for prime $q\neq p)$, or $|C(G)|=\frac{|G|}{2}+2^{k}, k\geq 0$. Then, we partially answer the open question by classifying finite groups $G$ having $\frac{|G|}{2}-1\leq |C(G)| \leq \frac{|G|}{2}+1$ for some fix values of $\mathcal{O}_{2}(G)$. Finally, we provide a complete list of finite groups $G$ having $|C(G)|=\frac{|G|+(2r+1)}{2}$ for $r\geq-1$.

math.GR↗
arXiv2024

On the Twisted Group Ring Isomorphism Problem for a class of groups

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for the associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for the groups of order $p^5$, where $p \geq 5$ is a prime, except a list of five groups.

math.GR↗
arXiv2023

On Projective representations of direct product of groups

Let $G=G_1 \times G_2$ be a finite group. We know that the second cohomology group $H^2(G,\mathbb C^\times)$ is isomorphic to $H^2(G_1,\mathbb C^\times) \times H^2(G_2,\mathbb C^\times) \times Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2', \mathbb C^\times ).$ A $2$-cocycle $α$ of $G$ is called a bilinear cocycle if the corresponding cohomology class $[α]$ of $H^2(G,\mathbb C^\times)$ lies in $Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2', \mathbb C^\times)$. In this article, our aim is to construct an irreducible complex projective representation $ρ$ of $G$ for bilinear cocycles $α$. If $G_1$ is any abelian $p$-group and $G_2$ is an elementary abelian $p$-group, then we give a construction of $ρ$ for bilinear cocycles $α$ of $G$. For a subgroup $H$ of $G$ of index $\leq p^2$, we also count the number of cohomology classes $[α]$ for which the irreducible projective representations behave the same while restricting on $H$. Finally, we consider any $p$-group $G=G_1\times G_2$, and we discuss how the above construction helps us to describe an irreducible $α$-representation of $G$ when $[α]$ is of order $p$ or $G_2/G_2'$ is elementary abelian. We also discuss several examples as an application of the above results.

math.RT↗
arXiv2022

An exact sequence and triviality of Bogomolov multiplier of groups

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology classes which vanish after restricting to every abelian subgroup of $G$. We give a new proof of a Hopf-type formula for $B_0(G)$ and derive an exact sequence for the cohomological version of the Bogomolov multiplier. Using this exact sequence we provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map. Finally, we give a complete list of groups of order $p^6$, for odd prime $p$, having trivial Bogomolov multiplier, so completing the 2020 investigation of Chen and Ma.

math.GR↗
arXiv2022

Projective representations of Heisenberg groups over the rings of order p^2

In this article we describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order $p^2$. We describe all projective representations of Heisenberg groups with entries from the rings $\mathbb Z/p^2\mathbb Z$ and $\mathbb{F}_p[t]/(t^2)$ and obtain a classification of their degenerate and non-degenerate 2-cocycles.

math.GR↗
arXiv2020

On Schur multiplier and projective representations of Heisenberg groups

In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations.

math.GR↗
arXiv2019

Schur multipliers of special p-groups of rank 2

A group G is called special p-group of rank k if the commutator subgroup [G,G] and centre Z(G) are equal, which is elementary abelian p-group of rank k and G/[G,G] is also elementary abelian p-group. In this article we determine the Schur multiplier of special p-groups of rank 2 explicitly.

math.GR↗
arXiv2018

The Schur Multipliers of $p$-Groups of Order $p^5$

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian $p$-groups of order $p^5$. As an application we determine the capability of groups of order $p^5$.

math.GR↗
arXiv2017

The Schur multiplier of central product of groups

Let $G$ be a central product of two groups $H$ and $K$. We study second cohomology group of $G$, having coefficients in a divisible abelian group $D$ with trivial $G$-action, in terms of the second cohomology groups of certain quotients of $H$ and $K$. In particular, for $D = \mathbb{C}^{*}$, some of our results provide a refinement of results from [Some groups with non-trivial multiplicators, Math. Z. {\bf 120 } (1971), 307-308] and [On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra {\bf 3} (1973), 73-82].

math.GR↗
arXiv2017

Classification of p-groups by their Schur multiplier

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. So $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. The groups has already been classified for $t(G) \leq 5$ by several authors. For $t(G)=6$ the classification has been done. In this paper we classify $p$-groups $G$ for $t(G) = 6$ in different method.

math.GR↗
arXiv2017

A characterization of finite $p$-groups by their Schur multiplier

Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of order $p^n$ for $t(G)=\log_p(|G|)+1$.

math.GR↗
arXiv2016

Finite $p$-groups having Schur multiplier of maximum order

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ denote the Schur multiplier of $G$. Niroomand proved that $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)+1}$ for non-abelian $p$-groups $G$ of order $p^n$ with derived subgroup of order $p^k$. Recently Rai classified $p$-groups $G$ of nilpotency class $2$ for which $|M(G)|$ attains this bound. In this article we show that there is no finite $p$-group $G$ of nilpotency class $c \geq 3$ for $p\neq3$ such that $|M(G)|$ attains this bound. Hence $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)}$ for $p$-groups $G$ of class $c \geq 3$ where $p \neq 3$. We also construct a $p$-group $G$ for $p=3$ such that $|M(G)|$ attains the Niroomand's bound.

math.GR↗