# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,874
questions

**5**

votes

**0**answers

71 views

### Intersection of $\mathrm{PGL}_2(q_0)$'s in $\mathrm{PGL}_2(q_0^3)$

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I would like an explanation for some strange numerology which I encountered when studying intersections of subfield subgroups in $\PGL_2(q)$. ...

**1**

vote

**1**answer

85 views

### The $\{2,3\}$-groups with a condition about $\mathbb{C}$-characters

Let $G$ be a $\{2,3\}$-group and $\lvert G\rvert=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define
$$
\nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\...

**13**

votes

**2**answers

420 views

### Global homological dimension of group rings

In all that follows, let $k$ be a field and $G$ be a finite group.
It is well-known that the order of $G$ is invertible in $k$ iff the group ring $k[G]$ is semisimple, which is equivalent, inter alia, ...

**3**

votes

**0**answers

49 views

### Classification of maximal point groups

Have the maximal (without finite proper overgroups of the same dimensionality) finite point groups been fully classified in any dimensionality of Euclidean space greater than 4?

**6**

votes

**1**answer

185 views

### Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?

Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...

**1**

vote

**1**answer

128 views

### Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer).
This invariant has been studied ...

**3**

votes

**1**answer

69 views

### Example of a primitive group of affine type and of twisted wreath product type

Good evening,
According to the O'Nan-Scott theorem, primitive finite group are classified into five classes: affine type, product type, almost simple type, diagonal type and twisted wreath product ...

**8**

votes

**1**answer

149 views

### On the coefficients that appear in finite groups of matrices with integer entries

Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...

**3**

votes

**1**answer

373 views

### Are there overwhelmingly more finite posets than finite groups? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...

**4**

votes

**1**answer

203 views

### Find unitary transformation between two sets of matrices that represent group generators

I have a set of matrices $A_i$ that represent the generators of a finite group within a certain basis, and $B_i$ represent the same operators in a different basis.
How can I find a unitary ...

**7**

votes

**1**answer

199 views

### Fusing conjugacy classes

Consider a finite group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$.
Question. Is there some finite overgroup of $G$ which fuses $H$ and $I$ into a single conjugacy class?...

**2**

votes

**1**answer

273 views

### Is the representation of finite simple groups fully understood?

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...

**2**

votes

**0**answers

62 views

### Terminology and notation for generated subgroups

I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...

**2**

votes

**0**answers

169 views

### “Moonshine” basics?

Having browsed recently a bit about "moonshine", which looks to me like some weird surrealist landscape, I wonder if: 1. sporadic groups could be seen as seemingly isolated special points of ...

**17**

votes

**1**answer

465 views

### The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...

**2**

votes

**0**answers

91 views

### Calculating the polynomials which are invariant under the action of a simple finite group

Let $G$ be a simple, finite group. In general, $G$ is not abelian.
Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...

**9**

votes

**7**answers

876 views

### Representations of products of symmetric groups

I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say
$$ S_{...

**6**

votes

**2**answers

454 views

### Finite subgroups of $\operatorname{U}(2)$

Famously, the finite subgroups of $\operatorname{SU}(2)$ admit an ADE classification.
Question. Is there a similar result for finite subgroups of $\operatorname{U}(2)$? Are they
classified? If this ...

**0**

votes

**0**answers

124 views

### Algorithm to compute automorphism group of a finite group

Is there an algorithm to compute automorphism group of a finite group?
GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...

**3**

votes

**1**answer

230 views

### What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?

I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
...

**0**

votes

**0**answers

95 views

### Is a transfer homomorphism surjective?

Let $G$ be finite group with minimal number of generators$d$, and all his proper subgroups have at most $d-1$ as minimal number of generators.
Fix a normal subgroup $N$ of $G$.
For all subgroups $H$ ...

**0**

votes

**0**answers

51 views

### Multiplication on a group of given cardinal and random permutations

Let $n$ be an integer, that we assume to be large (the order of magnitude for the motivation about the question is about 2^100).
For the purpose of random automatic program certification, I need to be ...

**10**

votes

**2**answers

655 views

### Finite simple groups all of whose Sylow subgroups of odd order are cyclic

Let $G$ be a nonabelian finite simple group all of whose Sylow subgroups of odd order are cyclic.
If we further assume that its Sylow $2$-subgroup is dihedral, then due to Suzuki, we know that $G\cong ...

**1**

vote

**0**answers

72 views

### A question concerning finite metacyclic groups

Consider a finite metacyclic group with presentation
$$G = \langle x,y \, |\, y^n=1, x^u=y^r, x^{-1}yx = y^k \rangle,$$ where $r \mid n$.
Is it true that if $G$ does not split (i.e. $G$ is a not a ...

**6**

votes

**2**answers

596 views

### Proofs of a character identity?

Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...

**0**

votes

**1**answer

152 views

### Faithful irreducible representations of the dihedral group $D_{2p}$, of dimension at most $p-1$

I am curious about the irreducible representations $\rho: D_{2p} \rightarrow GL_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two ...

**7**

votes

**0**answers

211 views

### Does there exist a nontrivial perfect group with a “locally commuting” presentation?

EDIT: I originally insisted that the perfect group in question be finite, however I now realize that I do not need this condition, only that the generators used in the presentation have finite order. ...

**7**

votes

**1**answer

225 views

### CFSG-free bound for the number of generators of a finite simple group

We know that every finite simple group can be generated by $2$ elements.
This (correct me if I'm wrong) was proved, as far as I know, by Steinberg (Steinberg, R. (1962). Generators for Simple Groups. ...

**10**

votes

**3**answers

530 views

### Realizability of a real representation using real cyclotomic coefficients

Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...

**3**

votes

**0**answers

45 views

### Classifying/enumerating vertex-transitive simplicial polytopes

I'm interested in understanding the class of simplicial polytopes in $\mathbb R^n$ whose Euclidean isometry group $G$ acts transitively on the vertices. These are examples that I know of:
simplicial ...

**5**

votes

**1**answer

211 views

### Embedding an icosahedron

A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...

**3**

votes

**0**answers

117 views

### Orders of finite 2-simple groups

Given that an $n$-simple group is a group isomorphic to the direct product of $n$ simple groups, can arbitrarily many nonisomorphic finite 2-simple groups share the same order?

**0**

votes

**1**answer

183 views

### Generators of $SL(n,\mathbb F_2)$? [closed]

Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...

**6**

votes

**1**answer

159 views

### Cohomology of finite $p$-groups over integers in local fields

Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).
In ...

**4**

votes

**1**answer

124 views

### Subgroups of finite simple groups $L(q^f)$ of Lie type normalized by $L(q)$

The following is a question asked to me these days by Gülin Ercan.
Let $G = L(q^f)$ be a finite simple group of Lie type,
and let $L(q) \cong H \le G$ be the group of fixed points
of the automorphisms ...

**1**

vote

**0**answers

83 views

### Actions of finite groups on compact symmetric spaces

I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit ...

**19**

votes

**6**answers

1k views

### Almost squared finite groups

Definition. A finite group $G$ is called squared (resp. almost squared) if there exists a subset $A\subseteq G$ such that $G=\{ab:a,b\in A\}$ and $|G|=|A|^2$ (resp. $|G|=|A|^2-1$). Such a set $A$ will ...

**4**

votes

**0**answers

224 views

### A big class of finite groups

During my researches, I've obtained a class of finite groups as follows.
Let $\mathcal{C}$ be the class of all finite groups $G$ such that for every factorization $|G|=ab$ there exists a subgroup $H\...

**5**

votes

**0**answers

353 views

### A lemma concerning conjugations and normal subgroups (related to a theorem of Frobenius)

In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of ...

**48**

votes

**3**answers

2k views

### Is each squared finite group trivial?

A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...

**14**

votes

**2**answers

421 views

### What is known about the structure of finite groups admitting an automorphism where all elements have “norm” one?

Let $G$ be a finite group admitting an automorphism $\sigma$ of prime order $p$. Define the norm map $N:G\rightarrow G$ with respect to $\sigma$ by $N(g)= g\sigma(g)\sigma^2(g)\dotsb\sigma^{p-1}(g)$.
...

**12**

votes

**1**answer

288 views

### Is every finite $d$-dimensional matrix group generated by $d$ elements?

The question is in the title. If $\Gamma\subset\mathrm{GL}(\Bbb R^d)$ is a finite matrix group, can it be generated by (at most) $d$ elements?
I suspect that this hope is too naive, but I have no ...

**3**

votes

**0**answers

62 views

### Relation between root subgroups and the root system in unitary groups

Consider a 4-dimensional non-degenerate unitary space over a field of order 4. It can be shown that there are 45 isotropic lines. For each such a line one can associate a unitary transvection and each ...

**9**

votes

**1**answer

309 views

### Groups with maximal element order 6

I'm dealing with finite groups $G$ in which the maximal order of an element is $6$.
With GAP I found out that for all groups with order $<1000$ the number of elements of order 6 $k := |\{x\in G : ...

**8**

votes

**2**answers

359 views

### On $p$-groups with abelian automorphism group

Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in ...

**18**

votes

**1**answer

749 views

### Groups with a unique lonely element

Does there exist a finite group $G$ of order greater than two containing a unique element $g$ such that
$$
g\notin\langle x\rangle
\hbox{ for all $x\in G\setminus\{g\}$ ?}
$$
Or we have another ...

**2**

votes

**1**answer

213 views

### Status of a conjecture of Thompson

Let $ S $ be a finite group. Denote by $\mathcal{B}_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of ...

**9**

votes

**2**answers

433 views

### Characterization of the family of simple groups PSL(2,q) by tensor multiplicity

Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...

**10**

votes

**2**answers

336 views

### Sequence of epimorphisms of residually finite groups stabilizes

Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually ...

**1**

vote

**0**answers

182 views

### About non-abelian finite simple groups

Let $S$ be a non-abelian finite simple group. Also let $\pi(n)$ be the set of prime divisors of a positive integer $n$. Is it true that $$\big|\pi(|\mathrm{Out}(S)|)-\pi(|S|)\big|<\big|\pi(|S|)\big|...