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7
votes
1answer
221 views

Counting conjugacy classes in simple groups of Lie type

Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split. ...
7
votes
1answer
459 views

Has this “pseudo-quotient” of groups been studied before?

Let $G$ be a (finite) group with subgroup $H$. Pick out (left) coset representatives: $x_1=1, x_2, \dots, x_\ell$ (so $x_1H=1H=H$). Now form an $\ell \times \ell$ table with rows and columns labeled ...
7
votes
1answer
289 views

Isomorphic simple groups

It is known that $SL_{4}(\mathbb{F}_2)\cong A_8$. Obviously, this is equivalent to the existence of a subgroup of $Sl_4(\mathbb{F}_2)$ of index $8$. How to find such a subgroup?
7
votes
0answers
227 views

Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
7
votes
0answers
105 views

Fusion pattern in a cyclic subgroup of order 8

Can a finite simple group $G$ have an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$? In other words, does this fusion ...
7
votes
0answers
214 views

Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005 Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G. Here, of course, we ...
7
votes
0answers
299 views

The maximal order of an element in orthogonal groups over finite fields of characteristic 2

Let $q$ be a power of $2$ and let $(V,Q)$ be a quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...
6
votes
2answers
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 $n$?
6
votes
4answers
605 views

How many finite simple groups of order $p+1$?

I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number. But they don't seem to fall into any classification - have these all been determined? Is the number of them even ...
6
votes
4answers
766 views

For a representation V of a finite group G, when is Hom(W, W⊗V) trivial for all irreps W?

This is probably really easy, but I just need someone to help me get mentally unstuck. As part of a description of the McKay correspondence, I want to show that if $G$ is a finite subgroup of $SU(2)$ ...
6
votes
2answers
490 views

Two groups acting on a set.

Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair ...
6
votes
1answer
244 views

The best upper bound for the number of involutions in a finite non-abelian simple group

Let $G$ be a finite non-abelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<|G|/3$ or $3t+1 \leq |G|$. Is this the best upper bound for the number of ...
6
votes
6answers
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 ...
6
votes
5answers
369 views

Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
6
votes
2answers
692 views

Subgroups of groups of Square-free order

If $G$ is a group of square-free order with at-least three prime factors, $|G|=p_1p_2....p_r$, $(2< p_i < p_{i+1})$, does $G$ contain a cyclic subgroup of composite order? (As groups of ...
6
votes
1answer
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 = ...
6
votes
2answers
331 views

Character table entries and sums of roots of unity

It is well-known that the entries of the character table of a finite group are sums of roots of unity. Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\infty]$, can I find a ...
6
votes
3answers
552 views

Exact sequences of permutational representations?

Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ ...
6
votes
1answer
205 views

Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...
6
votes
3answers
1k views

Subgroups of p-groups

If $G$ is a (non-abelian) $p$-group, $|G|=p^n$, $n>3$, then it is elementary that $G$ contains a (normal) abelian subgroup of order $p^2$. It is also true that $G$ necessarily contains a normal ...
6
votes
2answers
838 views

classification of small complete groups

This is "escalated" from stackexchange. So, $S_n$ for $n \ne 2,6$, $\text{Aut}(G)$ for $G$ non-cyclic simple, $\text{Hol}(C_p)$ for $p$ odd prime are well-known classes of complete group. What's ...
6
votes
1answer
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 $\geq 2$ ...
6
votes
3answers
473 views

Center and representations of finite group - how are related ?

If finite group G has a center how does it influence the representations of this group ? And vice versa - can we see somehow the center (or some of its properties) from representations (from ...
6
votes
2answers
478 views

Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
6
votes
2answers
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 ...
6
votes
1answer
419 views

Many p,q-Sylow subgroups

It is a fact that the symmetric groups have as many 2-Sylow subgroups as possible. More precisely, for all $n \geq 1$, the number of 2-Sylow subgroups in $S_n$ is exactly $n!/2^{\nu_2(n!)}$, which is ...
6
votes
2answers
434 views

Common Computations in Group Cohomology

Let G=A⋊B, where A and B are abelian, and of coprime order. It seems, from my computations (and correct me if I'm wrong), that Z1(Cp,Cq) is trivial, for p and q different primes. Meaning that ...
6
votes
1answer
501 views

Centers of Semidirect Products

The following question is for my own curiosity as I take some time to get reacquainted with group theory. Let G be a semi-direct product of the groups N and K with multiplication defined by the ...
6
votes
2answers
653 views

Linear algebra and regular orbits

If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ ...
6
votes
1answer
578 views

Character table for the affine group of Z/p^nZ

Initial caveat: the following question could probably be answered by Google, MathSciNet or my library, if I could find the right search terms or book... but I've not had any luck today, so I hope ...
6
votes
2answers
777 views

Maximum Order of elements in $GL(n,Z)$

Hi, I know that $\mathrm{GL}(n,\mathbb{Z})$ has an element of order $m$ iff $\Phi(m)\leq n$, where $\Phi(m) = \varphi(m)$ if $p_1^{\alpha_1}\neq 2$ or $m=2$, $\Phi(m) = \varphi(m)-1$ if ...
6
votes
2answers
472 views

How does one compute induced representations for modular representations?

The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its ...
6
votes
1answer
291 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?
6
votes
1answer
493 views

A sequence of finite groups

Question: does there exist a strictly ascending sequence of finite groups $G_0<G_1<G_2<\dots $ such that for every $i \in \mathbb{N}$ there is $a_i \in G_{i+3}$ and the following two ...
6
votes
1answer
423 views

Actions of finite permutation groups on hereditarily finite sets.

Model theorists have a lot to say about so-called definable imaginary elements of a structure. One way to formulate imaginaries is the following: Suppose $\mathcal{M}$ is a structure with universe ...
6
votes
1answer
217 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 ...
6
votes
1answer
192 views

Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
6
votes
0answers
201 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 ...
6
votes
0answers
152 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 $G$ ...
6
votes
0answers
185 views

Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...
6
votes
0answers
233 views

Recovering a group from its number of symmetric embeddings?

Fix a finite group $G$, and let $f_G:\mathbb{N}\rightarrow\mathbb{N}$ be defined by setting $f_G(n)$ to be the largest $m$ such that $S_n$ contains $m$ disjoint (pairwise-intersecting at 1) copies of ...
6
votes
0answers
311 views

Cyclic Sylow $p$-subgroup in finite simple groups

Hello, I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt. All finite simple groups have a cyclic Sylow $p$-subgroup for some $p$ I ...
6
votes
0answers
145 views

Finding a database of representations as matrices

Sorry if this would be more appropriate as a stackoverflow and not a mathoverflow question, but I think it's more likely to be known in this community. There are plenty of places on the internet or ...
6
votes
0answers
356 views

Jordan's Theorem on primitive permutation groups

The theorem I am referring to in the title is this: Theorem. If $p$ is a prime and $n$ is an integer with $n \geq p+3$, then the only primitive permutation groups on $n$ points containing a ...
6
votes
0answers
480 views

When is Hom(G, H) the same size as Hom(H, G)?

Let $G$ and $H$ be finite groups. Consider the ratio $$r_{G, H} \equiv {|Hom(G, H)| \over{|Hom(H,G)|}}$$ My question is When is $r_{G, H} = 1$? Can we characterize the pairs of groups $(G, H)$ ...
5
votes
5answers
710 views

Groups of order $p(p^2+1)/2$

It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$? Thanks for your answers
5
votes
4answers
2k views

Real representation of finite groups

I want to know if there is complete theory on real representation of finite groups. Say, given a finite group G, can we know all the injections from G to GL(n,R) for a particular n?
5
votes
7answers
531 views

Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the ...
5
votes
1answer
881 views

A question on the product of element orders of a finite group

Let $G$ be a finite group of order $n$ and $\psi(G)$ be the sum of element orders of $G$. Then $\psi(G)\leq\psi(C_n)$, where $C_n$ is the cyclic group of order $n$ (see "Sums of element orders in ...
5
votes
3answers
2k views

Finite groups with elements of order n

Consider a finite group where all elements have the same order $n$. What could be said about such groups? For $n=2$ it could be proved that such group is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^k$. ...