The finite-groups tag has no wiki summary.

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**1**answer

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### Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = ...

**2**

votes

**1**answer

176 views

### a question about the semidihedral group?

My question is simple:
If a group $G$ has the same character table with the semidihedral group $SD_{2n}$, are $G$ and $SD_{2n}$ isomorphic ?

**11**

votes

**1**answer

216 views

### Crossed modules and degree-3 group cohomology

It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, ...

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votes

**0**answers

160 views

### character table and subgroups

As we know, the normal subgroups can be found by inspection from the character table of a group G,my question is if all subgroups can be found by the character table of a group G, if not, then what ...

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**3**answers

340 views

### character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here:
Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in {\rm Irr}(G)$?
Can any one tell me ...

**6**

votes

**1**answer

257 views

### Centralizers of elements in general linear group over Z mod prime power

I would like to know for which elements $x$ in $G:=Gl_n(\mathbb{Z}/\ell^e\mathbb{Z})$ their centralizers $C_G(x):=\{ y \in G \mid xy=yx\}$ are abelian groups.
Here, $n$ is an integer $\geq 2$ ...

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votes

**0**answers

113 views

### Non left $k$-Engel elements in a nilpoent group always generate this group

Given a finite nilpotent group $G$ and let us denote by $L_n(G)$ the set of left $n$-Engel elements in $G$.
Assume that $n$ is the smallest positive integer such that $L_n(G)=G$.
Is it true that $G$ ...

**2**

votes

**1**answer

180 views

### On some endomorphisms of finite groups of odd order

Let $G$ be a group of odd order. It is known that if every central automophism of $G$ acts trivially on the center, then $G$ is purely non-abelain, this amounts to saying that every central ...

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**1**answer

236 views

### Generalizing the Notion of Nilpotent/Abelian/Cyclic Numbers

Let us call a number $n\in\mathbb{N}$ nilpotent if
$$n=p_1^{e_1}\cdots p_m^{e_m}$$
with $p_i^k\not\equiv 1\mod p_j$ for $i,j\in\{1,\ldots,m\}$ and $1\leqslant k\leqslant e_i$.
A cute theorem says ...

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votes

**0**answers

242 views

### Uniform proof that a finite reflection group is determined by its degrees?

Given a finite (real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated polynomial ...

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votes

**2**answers

140 views

### Decomposition of $G$-harmonic polynomials

Let $G$ be a finite group and let $H_G$ denote the $G$-harmonic polynomials. What is the structure of $H_G$ as a $G$-module? Is it isomorphic to the regular representation?

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**1**answer

126 views

### The Frattini subgroup of $D_{\infty}$ [closed]

Please hint me. $\phi(D_{\infty})?$ $\phi(G)$ is Frattini subgroup of $G$, intersection of all the maximal subgroup of $G$ and
$D_{\infty}=<x,y|x^2=y^2=1>$.

**2**

votes

**0**answers

329 views

### A question about finite groups (a weak version of the converse of Lagrange theorem) [closed]

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that
there exists a subgroup of $G$ with order $d$ or $n/d$?
Of course this does not hold in full generality. -- In ...

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**0**answers

228 views

### Outer group automorphisms preserving conjugacy classes of pairs of commuting elements

The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.
Let $G$ be a finite group. An outer automorphism $\alpha$ of ...

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votes

**1**answer

258 views

### Finite groups with few double cosets with respect to abelian subgroup

The following question is motivated by the study of certain tensor categories, namely integral near-group categories.
Let $G$ be a finite group and $H\subset G$ be a subgroup. Is it possible to give ...

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votes

**2**answers

342 views

### The number of conjugacy classes of the simple group PSL(2,q)

If $q=p^a$ , where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$ .

**3**

votes

**1**answer

260 views

### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...

**6**

votes

**4**answers

727 views

### Presentation of the Monster Group

I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other ...

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votes

**1**answer

436 views

### What natural numbers can be considered as the product of orders of elements of a finite (abelian) group

Problem A5 from putname's competition 2009 asks to prove that there is no finite abelian group such that the product of order of its elements is equal to $2^{2009}$. Starting from this problem, for ...

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votes

**1**answer

295 views

### Quotients of Hurwitz group

Since my question at Simplicity of infinite groups was not answered (well, at least, my second question), instead of trying to find the isomorhism type of those groups, I will instead try to find the ...

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**3**answers

560 views

### Dimensions and number of complex irreducible representations for SL3(Z/pZ)

This is my first post here :)
I have the following two related questions. While looking in Conway's ATLAS (the 1985 one) for $SL_3(Z/pZ)$, where he does the cases $p=2,3,5,7$, I saw that the ...

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vote

**1**answer

284 views

### Help understanding a group

I was experimenting with various presentations for groups, and I stumbled upon $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [aba,b]^6 \rangle$. I found that it has order 11741184, but the magma ...

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**1**answer

217 views

### A generalization of an old group problem [closed]

Here is an old exercise in group theory: (1) If $G$ is a group of order $2n$ with $n$ odd then $G$ is not simple and in fact $G$ has a normal subgroup of order $n$. I am going for one straight ...

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votes

**2**answers

544 views

### Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...

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**3**answers

504 views

### Is there a Cayley graph of a non-abelian group that is not isomorphic to any Cayley graph of any abelian group?

It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others.
In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...

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votes

**1**answer

408 views

### A question on the number of subgroups of symmetric groups

Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a ...

**2**

votes

**2**answers

321 views

### Automorphism group action leads to a “quotient graph”

Let $G$ be a simple (finite) graph. Consider the next natural equivalence relation $\sim$ on $V(G)$:
$u\sim v$ iff there exists and automorphism $\phi\in Aut(G)$, such that $\phi(u)=v$.
Define a new ...

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**1**answer

351 views

### Group theory conjecture on hurwitz groups

Conjecture: Let $p$ be a prime.
Then the group
$G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9, (([a,b]^4)b)^{2p} \rangle$
has a composition series of the form
${\rm PSL}(2,8) - {\rm Z}_p - ...

**3**

votes

**1**answer

257 views

### Index of $Z(G)$ in the centralizer of an element of a finite 2-group

Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denotes the conjugacy class of $x$ in $G$.
Is it true that $|C_G(x):Z(G)|=2$?
...

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162 views

### A constant associated to the character table of a finite group

Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}_{\rm Z}}$
Let $G$ ...

**5**

votes

**1**answer

161 views

### How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?

Let $G \leq {\rm S}_n$ be a finite permutation group, and let
$S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The growth ...

**9**

votes

**2**answers

298 views

### Extension of the Weyl dimension formula

Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...

**4**

votes

**1**answer

149 views

### Number of generators of the automorphism group of an abelian group

Let $G$ be a finite abelian $p$-group. What is known about the minimal number of generators of a $p$-sylow of $Aut(G)$? is it bounded in terms of $d(G)$ the minimal number of generators of $G$ (and ...

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**2**answers

598 views

### Orthogonal orthomorphisms of order 2

EDIT: There is an additional requirement that the composition of the orthomorphisms will also be order 2 (see my answer below).
A full proof is not needed, I will be happy with any argument which ...

**1**

vote

**1**answer

78 views

### The action of graph automorphism of finite symplectic group on maximal subgroups

Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...

**6**

votes

**1**answer

229 views

### Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...

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**3**answers

563 views

### Word evaluating to a group element and its inverse with different frequency

I'm supervising an undergraduate research project. Among other things, I've got the student to look at this paper of Gene Kopp and John Wiltshire-Gordon. This question arose from a missing complex ...

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**0**answers

228 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

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votes

**2**answers

682 views

### Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...

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**0**answers

89 views

### Soluble subgroups of 8-dimensional orthogonal groups over GF(4) transitive on nondegenerate 1-subspaces

Let $V$ be an $8$-dimensional vector space over $GF(4)$ equipped with a nondegenerate plus type quadratic form, $G$ be an almost simple group with socle $L=\Omega^+(V)$, and $H$ be a soluble subgroup ...

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votes

**1**answer

130 views

### Subgroup structure of orthogonal groups of small dimension over finite fields

How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...

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**1**answer

87 views

### Generalized system normalizer for insoluble finite groups

For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has:
$G = K C_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$)
$G \neq K C_G(K/L)$ and ...

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votes

**1**answer

192 views

### An upper bound for the maximal subgroups at fixed index?

Let us call a subgroup an injective homomorphism between groups.
I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone.
A subgroup $H \subset G$ is ...

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1k views

### Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:
Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset ...

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votes

**1**answer

186 views

### non-split extension and Schur multiplier

Let $G$ be a central extension of the group $K$ by the group $H$. If we know that this extension is non-split, is it true that the order of $K$ must divide the Schur multiplier of the group $H$?

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votes

**1**answer

600 views

### Finite groups with no elements of order $p^2q$

Let $G$ be a finite group. What can be said if $G$ has the following Property P: $G$ has no element of order $p^2q$ for any two distinct primes $p,q$?
In particular, which finite simple groups ...

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votes

**1**answer

124 views

### Embeddings of of quotient singularities

Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. ...

**12**

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**1**answer

251 views

### Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$

Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...

**7**

votes

**1**answer

303 views

### Solvability of finite groups of order coprime to 15 — proof without using CFSG?

Is the solvability of finite groups of order coprime to 15
essentially easier to prove than the entire Classification of Finite Simple Groups?

**0**

votes

**1**answer

212 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...