# Tagged Questions

The finite-groups tag has no wiki summary.

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**1**answer

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### 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

**1**answer

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

**1**answer

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?

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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, ...

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**0**answers

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 ...

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**0**answers

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 ...

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**0**answers

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**

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**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 $n$?

**6**

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**4**answers

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 ...

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votes

**4**answers

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 $SU(2)$ ...

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**2**answers

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 ...

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**1**answer

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

**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 ...

**6**

votes

**5**answers

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

**2**answers

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

**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 = ...

**6**

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**2**answers

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

**3**answers

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

**1**answer

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 ...

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votes

**3**answers

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

**2**answers

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

**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 $\geq 2$ ...

**6**

votes

**3**answers

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 ...

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votes

**2**answers

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

**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 ...

**6**

votes

**1**answer

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

**2**answers

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

**1**answer

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 ...

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votes

**2**answers

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

**1**answer

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 ...

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votes

**2**answers

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 ...

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**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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**0**answers

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 ...

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**0**answers

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 $G$ ...

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votes

**0**answers

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 ...

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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 ...

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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 ...

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votes

**0**answers

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 ...

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votes

**0**answers

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 ...

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**0**answers

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

**5**answers

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

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votes

**4**answers

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

**7**answers

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

**1**answer

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 ...

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votes

**3**answers

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 $(\mathbb{Z}/2\mathbb{Z})^k$.
...