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4
votes
1answer
170 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 $GL(n,q)$, ...
25
votes
2answers
885 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
2answers
337 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
1answer
245 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
1answer
149 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
1answer
633 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
4answers
461 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 ...
6
votes
2answers
259 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
2answers
353 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
1answer
200 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
1answer
201 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))
14
votes
1answer
593 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
1answer
93 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
2answers
469 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 $n$?
3
votes
2answers
302 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
0answers
402 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
1answer
112 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
1answer
155 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
0answers
132 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 ...
49
votes
4answers
2k 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 ...
9
votes
1answer
304 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
2answers
235 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
1answer
150 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
2answers
339 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
1answer
173 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
1answer
140 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 ...
18
votes
8answers
2k 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
4answers
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
1answer
836 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
0answers
89 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
1answer
259 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
1answer
110 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
1answer
215 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
1answer
458 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
1answer
182 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
1answer
228 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
0answers
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 ...
6
votes
3answers
344 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
1answer
262 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
0answers
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
1answer
184 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 ...
6
votes
1answer
248 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 ...
19
votes
0answers
248 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
2answers
141 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
1answer
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
0answers
368 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 ...
6
votes
0answers
237 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 ...
11
votes
1answer
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 ...
5
votes
2answers
355 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
1answer
271 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 ...