# Tagged Questions

The finite-groups tag has no wiki summary.

**13**

votes

**1**answer

275 views

### Does O'Nan-Scott depend on CFSG?

My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...

**3**

votes

**2**answers

266 views

### n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$.
(1) What is the explicit forms of its cohomology group (see my definition) in a ...

**6**

votes

**6**answers

418 views

### Almost uniquely generated groups

This is inspired by this question.
Does there exist an infinite finitely generated group having
(a) a unique
(b) finitely many
inclusion-minimal generating set(s) up to ...

**2**

votes

**1**answer

166 views

### Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is
known. A good account of such classification can be found for example
in this paper:
Cameron, Peter J. Finite permutation groups ...

**18**

votes

**2**answers

484 views

### Reference for the triple covering of A_6

I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...

**4**

votes

**1**answer

162 views

### Cyclic subgroups of GL(n,q)

Let $q$ be a prime power. It is well known that every Singer subgroup (subgroup of order $q^n-1$) in $GL(n,q)$ are conjugate. My question is: If $H$ is a cyclic subgroup of order $m$ in $GL(n,q)$, ...

**24**

votes

**2**answers

841 views

### A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...

**5**

votes

**2**answers

321 views

### Decomposing the conjugacy representation of Sym$(n)$ for small $n$

I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.
My own calculations are quite slow, ...

**11**

votes

**1**answer

234 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

**0**

votes

**1**answer

145 views

### relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free?
This question is related to my previous question
Relatively free algebras in a variety ...

**16**

votes

**1**answer

513 views

### Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...

**4**

votes

**4**answers

356 views

### Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...

**5**

votes

**2**answers

247 views

### Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...

**7**

votes

**2**answers

343 views

### How many finite loops?

How many finite loops of order $n$ are there?
I am interested in the exact values of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$.
Note ...

**1**

vote

**1**answer

166 views

### The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$

Given a finite Abelian group: $G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where ${n_1},{n_2},{n_3}$ are arbitrary positive integers. ${n_1},{n_2},{n_3}$ may have or may not ...

**-4**

votes

**1**answer

174 views

### Simple group of order 504 [closed]

As we know,there are 9 Sylow 2-subgroup in the Simple group of order 504.Can anyone prove it only by Sylow's theorem?
(you can't use knowledge about PSL(2,8))

**11**

votes

**1**answer

398 views

### Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function?
Background:
The counting function, $f(n)$, is a ...

**2**

votes

**1**answer

84 views

### Flag primitivity of the correlation group of classical projective planes.

We know that the full automorphism group of the $\pi_q = PG(2,q)$ acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full ...

**6**

votes

**2**answers

369 views

### Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?

**4**

votes

**2**answers

267 views

### Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,
$$G\times(X\times...\times ...

**9**

votes

**0**answers

380 views

### How many sporadic simple groups are there, really?

I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...

**1**

vote

**1**answer

101 views

### Specht polynomials for dihedral groups

The representation theory of the symmetric group is best understood via the Specht polynomials. In wonder how this works for other finite reflection groups, such as dihedral groups. Are the similarly ...

**2**

votes

**1**answer

149 views

### Simplification problem for finite groups

Let $G_1,G_2,H$ be finite groups. My question is:
if $G_1\times H$ is isomorphic to $G_2\times H$, is $G_1$ isomorphic to $G_2$?
I came to this question while preparing an exercise on finite abelian ...

**1**

vote

**0**answers

125 views

### A connection between nonplanar complete graphs and the alternating group?

I didn't get any response on MSE so I though I'd give this a try here (my question on MSE).
I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...

**41**

votes

**3**answers

1k views

### Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...

**8**

votes

**1**answer

294 views

### A subgroup intersects conjugacy class of every prime power order element

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that for any prime power order element $x$ of $G$, there exists some element $g$ in $G$ such that $x^g$ is contained in $H$. Does it ...

**7**

votes

**2**answers

187 views

### On non-split extensions of $\mathrm{SL}_d(q)$

Throughout this question, the following notation holds: Let $q$ be a power of a prime $p$, and let $d>4$ be a positive integer. Let $G$ be a finite group with a normal subgroup $E$ which is an ...

**0**

votes

**1**answer

144 views

### A finite $p$-group with certain properties

Is there a finite $p$-group $G$ such that :
(a) $G= \langle A,x,y \rangle$, with $G/Z(G)$ has exponent $p$, $A$ is a maximal abelian normal subgroup of $G$, and $G/A$ has order $p^2$ (thus it is ...

**6**

votes

**2**answers

328 views

### Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$

The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for ...

**3**

votes

**1**answer

141 views

### Involution centralizers in simple groups

I often see lower bounds on the size of centralizers of involutions
in finite (nonabelian) simple groups, but is there a general upper bound
for the size of an involution centralizer in such a ...

**5**

votes

**1**answer

135 views

### Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over ...

**15**

votes

**6**answers

1k views

### Fantastic properties of Z/2Z

Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...

**27**

votes

**4**answers

2k views

### For which $n$ is there only one group of order $n$?

Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts:
If $n$ is not squarefree, then there are multiple abelian groups of order $n$.
If $n \geq 4$ is ...

**20**

votes

**1**answer

792 views

### Monstrous moonshine for $M_{24}$ and K3?

An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...

**2**

votes

**0**answers

87 views

### Finite subgroups of compact simple Lie groups [duplicate]

The finite subgroups of $SU(2)$ consist of the symmetry groups of the Platonic solids plus the finite subgroups of $O(2)$. I would like to know if there are any results concerning $SU(3)$. In ...

**8**

votes

**1**answer

238 views

### A question about representations of finite groups

Let $G$ be a finite group and let $V$ be an irreducible complex representation of $G$.
Does there exist an element $g \in G$ which acts on $V$ with distinct eigenvalues?
If true, can you provide a ...

**0**

votes

**1**answer

106 views

### Subgroups of finite reflection groups that do not fix a point

Let $(W,S)$ be a finite irreducible Coxeter-System of rank $n$ and $E$ be a real reflection representation of $W$. Let $x\in E$ and suppose that the isotropy group of $x$ is generated by one element ...

**8**

votes

**1**answer

194 views

### Obstruction to extension of non-abelian groups - finite example?

Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner ...

**6**

votes

**1**answer

437 views

### 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

158 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

196 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, ...

**0**

votes

**0**answers

152 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 ...

**5**

votes

**2**answers

282 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

243 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 $\geq 2$ ...

**2**

votes

**0**answers

110 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 $G$ ...

**3**

votes

**1**answer

178 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 ...

**5**

votes

**1**answer

206 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 ...

**17**

votes

**0**answers

171 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 ...

**0**

votes

**2**answers

138 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?

**0**

votes

**1**answer

125 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
$D_{\infty}=<x,y|x^2=y^2=1>$.