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Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

4
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0answers
179 views

Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$. In fact, it's not important that I have exactly one representative from ...
3
votes
2answers
364 views

Finite groups with small God's numbers

Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word ...
0
votes
1answer
167 views

Is $G$ non-solvable?

Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that (1) there is a character $\chi\in\mathrm{...
5
votes
0answers
70 views

A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...
1
vote
0answers
90 views

Character degrees of a finite group?

Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
12
votes
1answer
277 views

Genus of Cayley graph of $A_5$ with two generators of order 5

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. I am wondering what is its graph genus (orientable or non-orientable). The best I could get by trial and error is an ...
4
votes
0answers
112 views

Does $G$ have a normal abelian Sylow $2$-subgroup?

Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...
3
votes
0answers
77 views

Minimal dimension of faithful representation of $G(\Bbb F_q)$

Let $k=\Bbb F_q$ be a finite field, $\mathcal{G}$ be a reductive group over $k$, denote $m(G)$ by the minimal dimension of faithful representation of $G=\mathcal{G}(\Bbb F_q)$. Do we know the value $m(...
7
votes
1answer
505 views

On the structure of a finite group of order $144$

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$. QUESTION: Prove $G\cong A_4\times A_4$. By using Magma, we know ...
4
votes
1answer
159 views

Kantor's Singer cycle theorem

I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq ...
8
votes
1answer
269 views

Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?

Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$. Question: Is $\theta$ extendible to $G$?
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votes
0answers
122 views

Absolute center of a certain $p$-group $G$

Let $L(G)$ denote the absolute center of group $G$ that is the subgroup of $G$ consisting of those elements of $G$ which are kept fixed by every automorphism of $G$. We know that $L(G)$ is contained ...
6
votes
1answer
182 views

How much do we need to add to the generating set of the symplectic group to get $SL(2n,2)$?

Here $Sp(2n,\mathbb{F}_2)$ means the group of matrices preserving the form $\Omega = \left( \begin{array}{cc} 0&I \\ -I&0& \end{array} \right)$, i.e. the symplectic group over an even ...
1
vote
0answers
66 views

The degrees of ordinary characters of $PSp(2n,q)$ and $P\Omega O(2n+1,q)$

The finite simple groups $PSp(2n,q)$ and $P\Omega O(2n+1,q)$ have a same order, where $n\geqslant3$ and $q$ is odd. What are the degrees of the ordinary characters of these two groups? Thanks!!!
1
vote
0answers
119 views

Counting conjugacy classes with a subgroup of prime index

I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a ...
5
votes
1answer
187 views

Schur covers of affine 2-transitive groups

I am interested in Schur covers of minimal 2-transitive groups. A theorem of Burnside gives that every finite 2-transitive group is either almost simple or affine. In the time since, these groups have ...
1
vote
0answers
62 views

p-group of maximal class

I am trying to prove that if $G$ is a $p$-group of maximal class and order $p^4$ ($p$ odd), then its unique two-step centraliser $G_1=C_G([G,G])$ is of the form $C_{p^2}\times C_p$. It is clear from ...
0
votes
0answers
73 views

Example of Chevalley group

I'm trying to grasp my head around Chevalley groups. I could greatly benefit from an illustrating example here. Could somebody give an example of constructing a simple but nontrivial Chevalley group? ...
26
votes
2answers
866 views

Is the cohomology ring of a finite group computable?

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
2
votes
1answer
122 views

Maximal subgroups of Alternating groups of degree $p$, for some prime $p$

Let $p\neq5$ be a prime number such that $q=(p-1)/2$ is prime. Does there exist an Alternating group of degree $p$ in which every minimal subgroup of order $p$ is contained properly in exactly one ...
3
votes
1answer
161 views

Computation of group homology $H_2 ((\mathbb{Z}/3\mathbb{Z}) \rtimes (\mathbb{Z}/4\mathbb{Z}),\mathbb{Z})$

In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we ...
2
votes
0answers
60 views

A lattice ordered by inclusion and isomorphic to the lattice of quotient groups of a finite group

Let $G$ be a finite group. Consider the lattice $$L=\{ G/N:\text{$ N $ is a normal subgroup of $G $}\},$$ where $G/N \leq G/K$ if and only if $K\leq N$. The lattice operations ∧ and ∨ on quotient ...
5
votes
0answers
104 views

$m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties: $(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
4
votes
1answer
114 views

Central extensions of Suzuki 2-groups

Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a solvable group of automorphisms permutes the involutions ...
5
votes
1answer
178 views

Large subgroups of $S_n$ without large symmetric or alternating subgroups

I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form. $G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their ...
1
vote
0answers
50 views

Relation Among Conjugacy Classes

This is more a request to find out if there is any work in the literature discussing certain things. Is there a naturally defined partial ordering on the set of conjugacy classes of a finite group G? ...
4
votes
0answers
246 views

How many conjugacy classes of elementary abelian subgroups of rank $2$ does $GL_{n}(Z / pZ)$ have?

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
6
votes
1answer
326 views

Group (Co)Homology of Symmetric Group

The question concerns the group homology or group cohomology of symmetric groups. The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$. groupprops....
3
votes
0answers
91 views

does infinite leinster groups indicates infinite perfect numbers?

The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
1
vote
1answer
178 views

A question on the number of involutions in a 2-group

Is there a classification of groups of order $2^n$ and exponent $4$ containing exactly $2^{n-1}-1$ involutions?
4
votes
1answer
92 views

cohomology of finite groups of lie type with coefficients in the adjoint module

Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with ...
1
vote
0answers
116 views

Conjugacy classes of non-normal subgroups of a finite $p$-group

Let $G$ be a finite $p$-group of derived length $d$ and nilpotency class $c$. Suppose that $G$ is not a Dedekind group (i.e., possesses at least one non-normal subgroup). Suppose that $G^{(d-1)}$ has ...
16
votes
1answer
717 views

Why do these two Monster-related calculations yield $163$?

Fact 1: (1979, Conway and Norton)$^{1}$ "There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster." Note: There are 194 (linear) irreducible ...
1
vote
0answers
87 views

$G$-invariant bilinear maps

Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
0
votes
0answers
29 views

What is an example of a constructive encoding of binary strings modulo an arbitrary permutation group $G$?

Given a group $G \leq S_n$ we can construct by the axiom of choice a function $f : \{0, 1\}^n \rightarrow \{0, 1\}^{ceil(log_2 |\{0, 1\}^n/G|)}$ such that for any orbit $O$ of binary strings under $G$,...
3
votes
1answer
213 views

Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
3
votes
2answers
183 views

Length of composition series in a primitive group

Let $G$ be a primitive group acting on a set $\Omega$ with $n$ elements. By Cameron/Liebeck (essentially a consequence of the Classification + O'Nan-Scott), there are two possibilities: (a) $G$ has a ...
3
votes
1answer
97 views

The nilpotency class and the derived length of a $p$-group

Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$. As is well known, we have $d\leq \lfloor\log_2 c\rfloor+1$, (https://groupprops.subwiki.org/wiki/...
4
votes
2answers
131 views

Factorizations in terms of characters

I have asked this in math.SE (https://math.stackexchange.com/questions/2698772/factorizations-in-terms-of-characters) but it was barely viewed. I have seen mention in different places that the number ...
12
votes
2answers
575 views

Graph automorphism group

Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...
1
vote
1answer
95 views

Sub-circle-free Christmas-gift-giving

Suppose there is a group of $n$ people giving gifts to one another. Everybody brings a gift but we want the gifts to be "well-distributed" in the group. By this I mean the following: In how many ...
11
votes
1answer
305 views

Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?

Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is: When does there exist a ...
5
votes
2answers
338 views

A finite group that splits and does not split

Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup) such that there exists an epimorphism $$\tau \colon ...
2
votes
0answers
70 views

Permutation factorizations according to number of generated orbits

Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$. How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
2
votes
0answers
51 views

Is it possible to characterize all finite groups $G$ whose coprime graph contains precisely three or precisely four leaves?

Is it possible to characterize all finite groups $G$ whose coprime graph contains precisely three or precisely four leaves? In section 3 of X. Ma, H. Wei, and L. Yang, The coprime graph of a group, ...
4
votes
1answer
161 views

Time Complexity of the Word Problem for Finite Permutation Groups

Given a finite permutation group, i.e. a subgroup of the symmetric group on $n$ symbols in terms of generators, what is the complexity of the word problem? That is, computing if two words in the ...
10
votes
2answers
210 views

For every prime divisor $p$ of a finite 2-generator group $G$, is there a generating pair containing an element of order divisible by $p$?

Let $G$ be a finite 2-generated group. Let $p$ be a prime dividing the order of $G$. Must there exist a generating pair $(g,h)$ of $G$ such that $|g|$ is divisible by $p$? If not, is this true at ...
1
vote
1answer
72 views

Complexity to decide for permutation group if every element fixed at most $k$ points

I want to consider the following problem, which generalises the decision problem to decide if a given finite permutation group is a Frobenius group: Given a finite permutation group in terms of its ...
1
vote
2answers
121 views

Complexity of decision problem to decide if permutation group is $k$-transitive

Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a ...
7
votes
2answers
198 views

Finitely generated group splitting non-trivially over an infinite virtually cyclic subgroup

Let $G$ be a finitely generated group splitting non-trivially over a infinite virtually cyclic subgroup $V$, namely an amalgamated free product $$ A*_V B,\; V \neq A,B. $$ Question: can the group $G$...