The finite-groups tag has no wiki summary.

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### 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 ...

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### Orders of automorphism groups of p-groups

There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} ...

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### Do subgroups have “two sided bases”?

Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 ...

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### A Realization Problem for Character Tables

Given an $n\times n$ table of complex numbers, are there known sufficient conditions for the table to be the character table of a finite group? Representation theory gives plenty of necessary ...

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### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

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### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

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### 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 ...

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### Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ?
(Main case - complex numbers, comments on other cases are also welcome. "Given" ...

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258 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 ...

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857 views

### Textbook source for finite group properties deducible from character table?

Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...

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### Reference for this theorem in representation theory?

Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of ...

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### Incompressible surfaces in an open subset of R^3

Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have:
$\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H_2(U;\mathbb Z)=\mathbb Z$).
...

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881 views

### What does the typical non-solvable group look like?

According to a result of Higman and Sims (which I learned about in this paper of Poonen's) the typical p-group is 3-step nilpotent of a particular form. In particular the typical group is a 3-step ...

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### 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 ...

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683 views

### Applications of fusion systems

What are the applications of theory of fusion systems to finite group theory
or representation theory of finite groups? More concretely, is there any important
result in finite group theory or ...

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### Which finite groups are generated by n involutions?

One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...

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### The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...

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209 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 ...

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367 views

### Decomposition of induced representations / Refinement of Mackey's criterion

There are already some questions with almost the same title, but they are more restrictive.
Let $G$ be a finite group, $H$ a subgroup, $V$ an irreducible representation of $H$,
and $W=Ind_H^G V$ the ...

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### When Cayley graphs of the symmetric group wrt generating sets of transpositions are isomorphic?

Dear All,
I thought the following question might be well-known, but couldn't find anywhere, so decided to ask here:
Let $A$ and $B$ be two generating sets for $S_n$, consisting of transpositions.
...

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344 views

### Radical of $F_p[SL(2,p)]$

Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F_p[G]$ such ...

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### An extension of the converse to Hall's theorem.

This is an extension of this MSE question, in which I asked whether there was a counterexample to the following statement,
Conjecture. If a finite group $G$ contains a $\lbrace p,q \rbrace$-Hall ...

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350 views

### Parker-like loop of order 2187?

The Parker loop has order $2^{13}$, and reducing it modulo its 'center' yields -- not just a $12$-dimensional $\mathbb{Z} /(2)$-vector space, but one which can be naturally identified with the ...

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### Groups with an automorphism of order two fixing only two elements

It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion.
Is anything known about ...

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### Reference for restriction of a simple module over a splitting field to a smaller field?

This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group ...

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### Cohomology of orthogonal and symplectic groups

Hello,
in their book Cohomology of Finite Groups Adem and Milgram investigate the cohomology of the finite orthogonal and symplectic groups only in case $\mathbb{F}_2$.
Let $p$ be a prime dividing ...

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### Number of faithful representations of a finite group

Is it known how many faithful linear representations a finite group G has on a complex vector space of given dimension? What if G is abelian?
I would even be interested in this special case: the ...

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360 views

### Homomorphic images of a Cartesian product of finite groups

What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...

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### Bounding the number of character degrees of a finite group in terms of the order of the group

Let $cd(G)$ be the set of degrees of irreducible complex characters of the finite group $G$ (so $cd(G) = \{\chi(1) | \chi\in Irr(G)\}$).
What bounds are known of the form $|cd(G)|\leq f(|G|)$ (ie, ...

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### Can one find the size of a Sylow normalizer from the character table?

Is the size of the normalizer of a Sylow p-subgroup determined by the ordinary character table of the group?
And if so, how does one calculate it?
In a solvable group, apparently one can compute ...

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### Periodic Automorphism Towers

In Scott's classic textbook on Group Theory, he asks:
Suppose that $G$ is a finite group. Is the sequence of isomorphism types
of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic?
...

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### Subgroups of $GL(k,q)$ for bounded $k$

This question on subgroups of $GL(2,q)$ asked by Jan, and especially wonderful answers to it given by Geoff Robinson, Ralph, and Will Sawin showing that "almost no finite groups" inject in $GL(2,q)$ ...

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### Aut(G) = $C_3$, G = ?

Is there a group G such that Aut(G) = $C_3$? What if we replace 3 with a prime number p?

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### Faithful characters of finite groups

Related to a previous question I am asking furthermore a proof
for the following:
Question 1: If $\chi$ is a faithful irreducible character of a
finite group $G$ then the regular character of $G$ is ...

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### 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 ...

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### What about the classification of big finite simple groups?

How hard is it to classify all big finite simple groups, i.e., all finite simple groups larger than some sufficiently large constant? Alternatively - how hard is it to classify all finite simple ...

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### How to compute all irreducible representations of a finite group ? (how GAP is doing this?)

Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever.
Question: How ...

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### Finite groups with centerless quotients

Is there a description of finite groups whose all quotients have trivial center? Is it true that only direct products of non-abelian simple groups have this property?

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### Is the classification of finite p-groups a smooth problem?

Fix a prime $p$. Then the question is about the difficulty of classifying finite $p$-groups. Originally I was going to ask if this was a "wild" problem but thanks to Joel David Hamkins' answer to ...

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### characters on a finite group with `extremal' behaviour

The following question is a bit technical, and I haven't got to grips with it enough to be able to present it as a well-focused question. However, my hope is that the collective group-theoretic ...

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### Orthogonal Groups over finite fields

Hello
Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms.
So here I want to pick
any non-degenerate ...

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### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

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### Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden :
See Here
He quoted the following claim:
"There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides
$ \Pi_{k=0}^{n-1} ...

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### What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...

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### Steinberg Representations of Finite Groups of Lie Type

Let G be a finite group of Lie type. Assume G is also of universal type. Is the Steinberg representation of G generic, i.e., does the Steinberg representation admit a Whittaker model?
A Whittaker ...

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### 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 ...

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### 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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### Are there any abelian 2 groups with Complete Holomorphs Other than $C^2_2$ and $C^4_2$?

The title says it. I'm reviewing some group theory concepts I haven't touched in quite a while, since I don't teach abstract algebra in my current position, and could not find the answer to this ...

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### Are vertex and edge-transitive graphs determined by their spectrum?

A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges.
The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the ...

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### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...