Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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5
votes
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
7answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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
3answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
6answers
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
0answers
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
0answers
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
3answers
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
2answers
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
2answers
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
0answers
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|...

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