# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

2,311
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### Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$

Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...

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### Embedding rank of finite groups and quotients

Let $G$ be a finite group, and $n$ a positive integer. It is not hard to check that the following are equivalent:
For every $g\in G\setminus\{1\}$ there is a subgroup $H\leq G$ with $|G/H|\leq n$ ...

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### Stem extensions and quotients of Schur covers

Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is
$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$
with the image of $A$ contained ...

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### A projectivity property in the category of groups

Let $F_r$ be the free group on $r$ generators, let $G$ and $H$ be finite groups, and let $F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$ be surjective homomorphisms. It is then easy to see that we can ...

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### Automorphisms of powers of finite simple groups

It is a theorem that a finite nontrivial group $G$ has no proper nontrivial characteristic subgroups if and only if $G \cong S^n$ where $S$ is simple and $n > 0$ is the number of copies of $S$ in a ...

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### Generators of a Coxeter group

Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-...

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### Are there any non-conjugation "extendible automorphisms" in the category of finite groups?

Let $\mathbf{Grp}$ be the category of groups. Given a subcategory $\mathscr{G}$ of $\mathbf{Grp}$ and $G\in\mathit{Ob}(\mathscr{G})$, a $\mathscr{G}$-extendible map on $G$ will here mean an assignment ...

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### Isomorphism classes of finite $\mathbb{N}$-groups

Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...

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### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...

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### Upper bound for Davenport constant $D(S_n)$

I am working on the Davenport constant $D(G)$ for $S_n$ and $A_n$, i.e., the minimal number $d$ such that every sequence (multiset) of $d$ elements contains some subsequence giving identity as a ...

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### Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect

In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...

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### Relationship between units and primitive characters 2

This is a follow up to this question.
Let $(R,+,\cdot)$ be a finite ring.
Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with ...

1
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1
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### Relationship between units of a ring and primitive characters of the ring under addition

Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.
My question is the following problem:
Given the dual group $\...

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1
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### Equivariant Smith normal form?

Let $F$ be a finitely generated free $\mathbb{Z}$-module on which the group $G$ of two elements acts via group homomorphisms. Let $F'$ be a $G$-invariant submodule. By Smith normal form we know that ...

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### Normality of the intersection between a Carter subgroup and the nilpotent residual of a solvable group G

In his book "Group Theory," Schenkman, in the proof of Theorem VII.4.a, states that in a finite solvable group $G$, the intersection of a Carter subgroup $C$ (i.e., a self-normalizing and ...

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### Structure of a single automorphism of a finite abelian p-group

A finite abelian $p$-group $H$ is homogenous when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group ...

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### Involution centralizers in $\mathrm{PCSO}^{+}(8,3)$

According to Table 4.5.1 of [1], there should be 10 classes of involutions of type "p" and "e" in $\operatorname{Aut}(K)$ where $K=K_a=P\Omega^{+}(8,3)$.
And Table 4.5.1 also gives ...

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### How good is approximation of distance function on the Cayley graph by "Fourier" basis coming from the irreducible representations?

Consider finite group $G$ , symmetric set of its elements $S$, construct a Cayley graph.
Consider $d(g)$ - word metric or distance on the Cayley graph from identity to $g$.
As any function on a group ...

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### Doubly transitive groups in which a one point stabilizer has a normal subgroup of even size

In 1972, Hering classified the finite doubly transitive permutation groups $(G,X)$ ($G$ acting faithfully on $X$) in which $G_x$, with $x \in X$, contains a normal subgroup $N_x$ of even order which ...

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### Invariant subgroups for integer matrix groups

The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the group $\mathbb{Z}$. To be more ...

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### Structure of elements of a finite group not contained in any conjugate of a proper subgroup

Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...

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### Schur cover of alternating groups

Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric ...

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### How to choose N policemen positions to catch a drunk driver in the most effective way (on a Cayley graph of a finite group)?

Consider a Cayley graph of some big finite group. Consider random walk on such a graph - think of it as drunk driver. Fix some number $N$ which is much smaller than group size.
Question 1: How to ...

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### Transitive groups with fixed-point free elements of prime power order

A well-known result of Fein, Kantor and Schacher says that if $G$ is a finite group which acts transitively on a set $X$, then $G$ contains an element of prime power order without fixed letters. ...

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### Finite p-group $G$ of exponent $>p$ with all elements outside $\Phi(G)$ of order $p$

Does there exist a finite $p$-group $G$ of exponent $>p$, such that $o(g)=p$ for all $g\in G\setminus\Phi(G)$?

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### Irreducible tensor product representations in finite simple groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...

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### A question about coprime automorphisms of profinite groups

Let $p$ a prime. A finite group is a $p'$-group if its order is prime to $p$. Let $A$ be a finite $p'$-group of automorphisms of a finite $p$-group $G$. Suppose that $A$ is a non-cyclic abelian group. ...

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### Sylow subgroups of doubly transitive groups

Let $(G,X)$ be a doubly transitive permutation group (where $G$ acts faithfully on the set $X$). Let $x \in X$, and suppose that $\vert X \vert = n + 1$ is finite. Now let $p$ be a prime divisor of $n$...

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### Cardinal of finite orthogonal groups

Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.
By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...

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2
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### Generators of a group and normal subgroups

Can we say anything about a minimal generating set of a finite group based on its normal subgroups? For example, can we bound their order, or say whether they come from the same conjugacy class?
An ...

0
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1
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### Hyperoctahedral group, preliminaries [closed]

I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...

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### On Suzuki 2-groups being 2-Sylow subgroups

$\DeclareMathOperator\Sz{Sz}\DeclareMathOperator\SU{SU}$Recall the definition of the finite Suzuki 2-groups: These are finite non-abelian 2-groups that contain more than one involution such that a ...

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### Doubly transitive groups in which a point stabilizer has an abelian normal subgroup

Let $G$ be a finite doubly transitive group in its action on the set $X$, such that a point stabilizer $G_x$ ($x \in X$) has an abelian normal subgroup $N_x$.
I have read that if $\vert N_x \vert$ is ...

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### Maximal subgroups of finite abelian $2$-groups

Suppose $G$ is a finite abelian $2$-group, and $S$ is a subset of $G$, $\langle S\rangle=G$,$S^{-1}=S$,$e\notin S$. How to determine whether there exists a maximal subgroup $M$ of $G$, such that $S$ ...

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### Necessary and sufficient conditions for the Cayley graph to be bipartite

Let $G$ be a finite group with identity $1$. If $S$ be an inverse closed generated subset of $G$, then $S$ is called a Cayley subset of $G$.The Cayley graph $\Gamma=\operatorname{Cay}(G, S)$ is a ...

0
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### Commutator group and conjugacy classes

Let $G$ be a finite solvable group which is not nilpotent, and let $H=[G,G]$ be the commutator subgroup of $G$. Does the following hold for $G$ and $H$?
"There exists $g \in G \setminus H$ and $h ...

4
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2
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### Groups whose derived length is logarithmic in the order?

Is there a class of solvable groups $G$ having a derived length $O(\log\lvert G\rvert)$?
See Wikipedia for the definition of Big-Oh ($O$) and the definition of derived series of a group.
Any help ...

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### Largest primitive subgroup of $\mathrm{GL}_8(\mathbb{C})$ of order $2^a 3^b 5^c$

The paper "Bounds for finite primitive complex linear groups" by M. Collins computes the largest possible value of $[G:Z(G)]$ for $G$ a primitive subgroup of $\mathrm{GL}_N(\mathbb{C})$, for ...

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### Number of finite groups: is $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$?

Let $\operatorname{gnu}(n)$ be the number of finite groups of order $n$.
Question: Is it true that $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$ for all $n \geq 1$?
Surely this must be true, ...

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### What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$

In Table 22.1 on Page 193 of Malle & Testerman's book "Linear algebraic groups and finite groups of Lie type", the fixed point subgroup $G^F$ (where $F$ is a Steinberg endomorphism) of ...

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### Complex reflection groups: reference request

Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset \operatorname{GL}(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]...

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### Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$

$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...

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### Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request

Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below.
There can be several approaches to that task. One of ideas coming to my mind - in some ...

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### Shapiro's lemma for group derivations [duplicate]

Let $G$ be a finite group, let $H$ be a subgroup of $G$, let $W$ be an $H$-module and let $V$ be the $G$-module induced from $W$. Shapiro's lemma says that
$H^1(G,V)\cong H^1(H,W)$.
I was wondering if ...

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### The number of irreducible characters of simple groups of Lie type

Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$.
Set $\mathrm{C}_{S}(\sigma)$ the ...

6
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1
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### Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?

I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24:
Now let us recall that the ...

3
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1
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### Holt's Theorem on doubly transitive groups with $2$-central involutions fixing only one letter

In D. F. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, Proc. London Math. Soc. 37 (1978), 165–192, Derek Holt classifies finite doubly ...

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### Number of conjugacy classes of pairs of commuting elements II

This post follows up on a discussion initiated in Number of conjugacy classes of pairs of commuting elements.
Consider a finite group $G$ and let $r_G$ represent the number of conjugacy classes of ...

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### What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....

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### Groups $P$ of order $p^5$ with $\Omega_1(P)=P$

I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...