Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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1 vote
1 answer
135 views

Orbit sizes of $G=\operatorname{SO}^{+}_{2n}(2)$

Let $G=\operatorname{SO}^{+}_{2n}(2)$. I did some Magma computation and found there were $3$ orbits on the natural $G$-set when $n=2,3,4$. The orbit sizes are $1$-$9$-$6$, $1$-$35$-$28$, $1$-$135$-$...
-1 votes
1 answer
203 views

Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
3 votes
1 answer
230 views

Is there anything known about the lower central series of a group $G\wr C_p$?

Let $G$ be a $p$-group contained in $S_{p^m}$ for some $m\in \mathbb{N}$, this is, the symmetric group of degree $p^m$. Assume that the lower central series, LCS for short, of $G$ is well understood. ...
6 votes
1 answer
540 views

If G is an almost simple group, then Aut(G) is complete?

If G is an almost simple group, then Aut(G) is complete? Apologies - I meant to post this on Stack Exchange Just wondering if anyone has a reference to the above - it's quoted on Wikipedia (so ...
9 votes
2 answers
662 views

Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$

Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers? I can only find ...
2 votes
0 answers
149 views

The maximal subgroups of a finite solvable group

$\DeclareMathOperator\Syl{Syl}$$G$ is a finite solvable group and $G=Q\rtimes(P\times R)$, where $P\in \Syl_{p}(G)$ with $P$ is cyclic, $Q\in \Syl_{q}(G)$ with $Q$ is normal elementary abelian, $R\in \...
2 votes
1 answer
128 views

The property of subgroups of a finite solvable group

$\DeclareMathOperator\Syl{Syl}$$G$ is a finite solvable group and $G=PQR$, where $P\in \Syl_{p}(G)$, $Q\in \Syl_{q}(G)$, $R\in \Syl_{2}(G)$ and $|R|=2$. Suppose that $C_P(R)=P$ and $C_Q(R)=1$. Since $...
22 votes
0 answers
1k views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
8 votes
1 answer
445 views

I M Isaacs Algebra Exercise 9.4

I am a PhD student in the represention theory of finite groups. One of my friends and I solved all exercises in the book I M Isaacs - Algebra A Graduate Course except for the following exercise in ...
15 votes
1 answer
697 views

Finite abelian groups with fewer automorphisms than a subgroup

It is not too hard to find examples of finite groups which have fewer automorphisms than one of their subgroups. For example $\mathcal D_4 \times \mathbb Z/2\mathbb Z$ (where $\mathcal D_4$ is the ...
2 votes
0 answers
363 views

Conceptual proof of fundamental theorem of finite abelian groups

I'm looking for a conceptual proof of the following statement: Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
1 vote
0 answers
62 views

Distinguish $p'$-elements in a coset

Let $G$ be a finite group with a nonabelian minimal normal subgroup $N$. Then $N$ is a direct product of $n$ copies of some nonabelian simple group $S$. Let $p$ be a prime divisor of $|S|$ and let $xN\...
19 votes
3 answers
1k views

Can nonabelian groups be detected "locally"?

Suppose $m,n\geq 2$ are two integers. Is it true that for every sufficiently large nonabelian group $G$, one can find a set $A\subset G$, with $|A|=n$, so that $|A^m| >\binom{n+m-1}{m}$? (Edit) ...
3 votes
1 answer
328 views

Find an example where a subset of “inverse fixed points“ is not a subgroup

$G$ is a group of odd order, $\sigma$ is an automorphism of $G$, and $\sigma^2=\mathrm{id}$. I want to find an example to show that $G_s= \{ g \in G \mid \sigma \left( g \right)= g^{-1} \} $ might not ...
42 votes
3 answers
2k views

Are there "real" vs. "quaternionic" conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a symmetric form, self-dual by a symplectic form, and not self-dual at all. In the first two cases, the character is real-valued, ...
2 votes
0 answers
94 views

Number of prime factors of the order of an increasing sequence of finite non-abelian simple group

I recently come across this question: Number of prime factors of the order of a finite non-abelian simple group. What caught my attention is the second question: Does there exist a sequence $\{S_n\}$ ...
47 votes
1 answer
3k views

Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?

Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right? In other words, do there exist ...
2 votes
1 answer
112 views

If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$

Let $n=2m$. What is the order of the following permutation $\sigma$? $$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$
1 vote
0 answers
72 views

Bottleneck edge in lattice of subgroups

Let $G$ be a finite group. Define the bottleneck weight of a chain of subgroups $$\operatorname{id}=H_0 < H_1 < \ldots < H_n = G$$ to be the maximum value over the indices $[H_{i+1} : H_i]$ ...
2 votes
1 answer
153 views

Bound on the size of group related to a matrix basis

Let $ G $ be a group of $ n \times n $ matrices. Suppose that some subset $ \{ g_j: 1 \leq j \leq n^2 \} $ of $ G $ is a basis for the space of all $ n \times n $ matrices. Furthermore suppose that ...
4 votes
0 answers
188 views

Almost conjugate subgroups of compact simple Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group. Definition: Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
4 votes
1 answer
181 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?
5 votes
0 answers
134 views

The orders of which nonabelian finite simple groups can be written as products of other such orders?

Is it true that the order of a nonabelian finite simple group $G$ can be written as the product of the orders of two or more other nonabelian finite simple groups if and only if $G$ is either an ...
1 vote
0 answers
113 views

Is the commutator of the holomorph of generalized quaternion group abelian?

Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$. Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
35 votes
6 answers
5k views

Character-free proof that Frobenius kernel is a normal subgroup?

The question is in the title, but here is some background/reminders: A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
4 votes
4 answers
471 views

What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
3 votes
0 answers
249 views

Commuting real elements in finite groups

Let $p$, $q$, $r$ be three distinct odd primes, and $G$ a finite group with $|G|$ divisible by $p$, $q$, $r$ to the first power only. Let $x,y,z \in G$ be of order $p,q,r$ respectively. Assume (a.) $[...
4 votes
0 answers
129 views

Derived subgroup of rational points vs. rational points of derived subgroups

Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion $$ f: [G(k), G(k)] \rightarrow [G,G](k). $$ If $k$ is not algebraically closed, $f$ is not necessarily ...
4 votes
0 answers
133 views

Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$

What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?
5 votes
0 answers
114 views

Rigid points of $\mathrm{Co}_0$

Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups. How many conjugacy classes of rigid points are there under the ...
3 votes
1 answer
144 views

Bounds for the number of edges in an Alperin diagram

If $A$ is an algebra over a field $k$ and $M$ is a finite-dimensional $A$-module, then Alperin showed in a paper [Diagrams for modules, JPAA, 1980] how to associate a diagram to $M$ with the vertices ...
8 votes
0 answers
212 views

Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
10 votes
8 answers
1k views

Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "...
5 votes
1 answer
179 views

Characteristic subgroups of a finite abelian $2$-group

I have recently stumbled across the problem of describing the characteristic subgroups of a finite abelian group. With some discussions with some mathematicians in my lab, I managed to obtain a "...
2 votes
1 answer
198 views

Finite quotients of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime and let $ \SL^1_2(\mathbb{Z}_p)$ denote the kernel of the natrual surjective morphism $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p\mathbb{...
2 votes
0 answers
192 views

Characters of alternating groups

I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are: A character of dimension $3.696$ of $A_{...
40 votes
6 answers
5k views

What are some interesting corollaries of the classification of finite simple groups?

The classification of finite simple groups, whether it be viewed as finished, or as a work in progress, is (or will be) without doubt an enormous achievement. It clearly sheds a great deal of light on ...
25 votes
3 answers
5k views

learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups (...
7 votes
1 answer
281 views

Lovasz's conjecture for dihedral Cayley graphs

Background: A tantalizing conjecture of Lovasz is the following: Let $G$ be a (finite) connected vertex-transitive graph. Then $G$ contains a Hamiltonian cycle or is one of $5$ counter-examples. (...
3 votes
0 answers
115 views

Ways to tell from residues modulo prime factors if $z$ is below half point

Let $N=\prod_{k=0}^{k=m}{ p_k }$ be a square-free odd integer where $p_k$ is a prime. If we are given any integer $g$ such that $0<g<N$, it is very easy to tell if $g < \frac{N}{2}$ or not. ...
19 votes
1 answer
1k views

What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...
6 votes
0 answers
115 views

What are Burnside's "fixed systems" in modern language?

I just read Chapter 17, on rational invariants, in Burnside's classic 1911 text Theory of Groups of Finite Order. It was a great read, and mostly it was straightforward to translate the ideas into ...
2 votes
0 answers
60 views

Dual representation of a transitive semilinear group

This seems like it should be known, but I couldn't find a reference. Let $V$ be a finite vector space and let $G$ be a group of semilinear maps on $V$ (i.e. linear composed with a field automorphism). ...
6 votes
1 answer
533 views

On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
1 vote
1 answer
269 views

The property of self-normalizing subgroup

$G$ is a finite solvable group. Let $\{P_{1}, P_{2}, \dotsc , P_{s}\}$ be a Sylow basis of $G$. We have that $G=P_{1}P_{2}\dotsm P_{s}$. Set \begin{equation} \begin{aligned} %% The alignment is ...
0 votes
0 answers
80 views

Commutator magma isomorphism

Define the commutator magma of a group to be the magma whose elements are the same as the group’s and whose operation is the group’s commutator. What are the conditions for two finite groups to have ...
5 votes
1 answer
144 views

How to classify homomorphisms from $\operatorname{PSL}(2,p)$ to $\operatorname{PGL}(n,2)$ when $2^n=p+1$?

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I am looking to classify the homomorphisms from the group $\PSL(2,p)=\Aut(\mathbb{P}^...
3 votes
1 answer
193 views

Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...
2 votes
0 answers
87 views

Suzuki-Ree Lie algebras

Do the Suzuki and Ree groups of Lie type have associated Lie algebras over finite fields in the same way that the other groups of Lie type do? These algebras would be 5-dimensional over $\mathbb{F}_{2^...
5 votes
1 answer
413 views

Classification of natural endomorphisms on finite groups

Any $z \in \widehat{\mathbb{Z}} = \lim_{n} \mathbb{Z}/n\mathbb{Z}$ defines an operation on all finite groups: if $G$ is a finite group and $g \in G$, say $g^n=1$, then map it to $g^{z_n}$. This ...

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