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1
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1answer
117 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
157 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
134 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 ...
50
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
306 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
255 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
155 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
346 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
197 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
144 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 ...
28
votes
14answers
3k 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
861 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
265 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
111 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
221 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
476 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
200 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
242 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
169 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
351 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
281 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
115 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
186 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
263 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
255 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
145 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
127 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
377 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
248 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
394 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
279 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
4answers
792 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 ...
4
votes
1answer
439 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 ...
2
votes
1answer
304 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 ...
11
votes
3answers
588 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 ...
1
vote
1answer
290 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 ...
-2
votes
1answer
218 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 ...
8
votes
2answers
664 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 ...
7
votes
3answers
528 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 ...
-4
votes
1answer
435 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
2answers
358 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 ...
11
votes
1answer
379 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
1answer
270 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 $|C_G(x):Z(G)|=2$? ...
6
votes
0answers
167 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
1answer
172 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
2answers
311 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
1answer
151 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 ...