The finite-groups tag has no wiki summary.

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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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### 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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### 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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### 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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### 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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### 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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### Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation.
That is, a finite group $G$ is a Frobenius complement if and only ...

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### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

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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 ...

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### 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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### Is there order to the number of groups of different orders?

I was always struck by how uncharacteristically erratic the behavior of the following function is:
$f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(n):=$ number of isomorphism classes of groups of ...

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### Condition for a certain subset being a subgroup

For any finite group $G$ and $n$ a divisor of $|G|$, consider the following subset of elements of "co-order" dividing $n$:
$$G(n) = \{ g \in G \mid g^{|G|/n} = 1 \}$$
By a classical theorem of ...

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### 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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### Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...

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### 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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### 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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### Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...

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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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### 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 ...

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### 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 ...

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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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### 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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### What are the outer automorphisms of a Coxeter group?

I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.
But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter ...

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### Subgroup property stronger than being characteristic

In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...

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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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### 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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### Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
Question: For what groups does there exist a Hamiltonian ...

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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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### 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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### 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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### 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 ...

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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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### 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 ...

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### Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...

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### 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 ...

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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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### Burnside's Lemma and Geometry

I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action.
I wonder if there is any ...

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### Sylow subgroups invariant under an automorphism

Let $G$ be a finite group and $\sigma$ an automorphism of $G$. Suppose $p$ is a prime and $\sigma$ has prime order $q \neq p$. It's easy to see that $\sigma$ fixes a Sylow $p$-subgroup of $G$ if ...

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### 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 ...