Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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3 votes
1 answer
77 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^{...
92 votes
15 answers
32k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
1 vote
0 answers
77 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 ...
2 votes
0 answers
64 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
368 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 ...
0 votes
0 answers
86 views

$G=HK$, $N_H(P_4)=P_4$ and $N_K(P_4)=P_4$

$G$ is a solvable group. Let $\pi(G)=\{p_{1}, p_{2}, p_{3}, p_{4}\}$ and $\{P_{1}, P_{2}, P_{3}, P_{4}\}$ be a Sylow basis of $G$. We have that $G=P_{1}P_{2}P_{3}P_{4}$. Set $H=P_{1}P_{2}P_{4}$ and $K=...
19 votes
3 answers
4k 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
166 views

A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...
1 vote
0 answers
135 views

Character extension about $Q_8$

Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise: (Exercise 5.9) Let $G$ be a finite group and $N\unlhd G$, suppose ...
-2 votes
0 answers
127 views

Intersection of a Sylow subgroup with the center of its normaliser [closed]

Let $G$ be a finite group and $P \in \operatorname{Syl}_{p}(G)$. Let $z \in P \cap Z(N_{G}(P))$ be such that $tzt^{-1} \in P$, for some $t \in G$. Show that $tzt^{-1} = z$. I have tried a lot solving ...
26 votes
3 answers
2k views

Highly transitive groups (without assuming the classification of finite simple groups)

What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete ...
39 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 ...
4 votes
2 answers
299 views

Outer automorphism of a finite simple group which is isomorphic to a subgroup of $S_p$

Here is a statement having a proof that involved the CFSG. Let $p$ be a prime, and $S$ be a nonabelian finite simple group such that $S$ is isomorphic to a subgroup of $S_p$ with $p\mid |S|$. Then $\...
7 votes
1 answer
283 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. ...
6 votes
0 answers
258 views

(CFSG-free) Finite simple groups whose character degrees square divide its order

Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
-3 votes
1 answer
113 views

Exponential order of unipotent elements in an endomorphism ring of abelian groups

$\DeclareMathOperator\End{End}\newcommand{\Id}{\mathrm{Id}}$Let $E=\End(I)$ be the endomorphism ring of the abelian group $I$. We have the following statement for $B\in E$, $p$ a prime number and $r$ ...
0 votes
1 answer
149 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
215 views

Characters of tori in finite reductive group

Let $G$ be a connected split reductive group over a finite field $k$. Suppose $G$ has connected centre. Let $T$ be a maximal split torus with Weyl group $W$. Note that $W$ acts on the finite group $T(...
1 vote
1 answer
188 views

Kronecker product preserves the conjugacy relation?

Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-...
6 votes
2 answers
321 views

The action of a subgroup of the torsion group of elliptic curves on integral points?

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a ...
1 vote
0 answers
207 views

Presentation complexes with same homology and different fundamental groups

If we start with a perfect group $G$ of deficiency zero then there is a presentation $P$ of $G$ such that the number of relations and the number of generators for $P$ are the same. For such $P$, the ...
4 votes
1 answer
209 views

Does a perfect $4^{11}\cdot M_{24}$ exist?

Is there any perfect group which could be notated as $4^{11}\cdot M_{24}$ (a non-split extension of the largest Mathieu group by a homocyclic group of type $4^{11}$)?
1 vote
0 answers
98 views

Embedding (Kronecker product) preserves the structure?

In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix} -I_{i} & 0\\ 0 & I_{n-i} \end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
3 votes
0 answers
45 views

Hamilton cycles in Cayley graphs: between Rapaport-Strasser and Fleischner

A well-known question of Rapaport-Strasser asks whether every finite connected Cayley graph has a Hamilton cycle. Fleischner's Theorem implies that if $S$ is the generating set of such a Cayley graph $...
2 votes
1 answer
295 views

What are the maximal closed subgroups of $ \operatorname{SU}_3 $?

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU_3 $? This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that ...
1 vote
0 answers
103 views

Homogeneity of finite simple groups

According to A complete classification of finite homogeneous groups, the finite simple groups $G$ which are homogeneous (meaning every isomorphism between subgroups extends to an automorphism of $G$) ...
7 votes
0 answers
191 views

Locally finite groups containing all finite groups

Say that a group is rich if is contains isomorphic copies of all finite groups. It is easy to produce rich groups, and also rich locally finite groups, for instance the restricted direct product $A=\...
0 votes
1 answer
109 views

Intersection of identity components

Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
2 votes
1 answer
164 views

Invariants of the group algebra of a finite group

Consider a finite group $G$ and its complex group algebra $V_G$, on which $G$ acts. I would like to know: what are the polynomial $G$-invariants of $V_G$ i.e., the polynomial functions $p\in \mathbb{C}...
11 votes
2 answers
306 views

Minimal irrep of $\mathrm{PSL}(2,p) $

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ p $ be a prime for which $ \PSL(2,p) $ is simple (so $ p \neq 2,3 $). Is the ...
1 vote
2 answers
351 views

Admissible finite groups

(added) Definition. A complete map of a group is a permutation $\phi$ such that $g\mapsto g\phi(g)$ is also a permutation. A group is admissible if it admits a complete map. I want to know when an ...
1 vote
0 answers
50 views

Centralisers of involutions not quasi-isolated

The quasi-isolated elements of the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$ are classified in the paper "quasi-isolated elements in reductive groups" by C. Bonnafe. Let's focus ...
2 votes
1 answer
149 views

Fusing the $\mathrm{PGL}(2,11)$ conjugacy classes of $\mathrm{Aut}(M_{12})$

Is there an embedding of $\mathrm{Aut}(M_{12})$ into the automorphism group of some larger sporadic group that fuses its two conjugacy classes of $\mathrm{PGL}(2,11)$ subgroups?
4 votes
1 answer
190 views

Wedderburn decomposition of special linear groups

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
4 votes
2 answers
352 views

What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?

An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia. I ...
5 votes
1 answer
417 views

Finite maximal closed subgroups of Lie groups

Cross-posted from MSE https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups $\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{...
4 votes
4 answers
3k views

Automorphisms of non-abelian groups of order $ p^3$

There are two non-abelian groups of order $p^3$, namely, semi-direct product of $\mathbb Z /p \mathbb Z \times \mathbb Z /p \mathbb Z$ by $\mathbb Z /p \mathbb Z$ and semi-direct product of $\mathbb Z ...
0 votes
0 answers
74 views

A decision problem of an inverse problem in finite group theory

A finite group $G$ is called integral if there is a finite group $H$ such that $G\cong H'$. In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following:...
4 votes
1 answer
296 views

Converse of Clifford's theorem for a semidirect product

Suppose that a group $G$ is a semidirect product $G = N \rtimes H$ with $N \trianglelefteq G$. Let $\mathbb{F}$ be a field. Say $V$ is a finite-dimensional $\mathbb{F}[G]$-module such that $V \...
0 votes
0 answers
151 views

Signed permutations and $ \operatorname{SO}(n) $

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of determinant-$1$ signed permutations has order $ n!2^n/2 $. ...
1 vote
1 answer
80 views

Cohomological variety in case that Sylow subgroup is elementary abelian

Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V_G$ of ...
2 votes
1 answer
481 views

Is the dihedral group admissible?

I want to ask: When is the dihedral group $D_n$ admissible? Or, when does the Latin square of the Cayley table of a dihedral group have a transversal? thanks
18 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 ...
2 votes
0 answers
57 views

Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group

Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...
1 vote
0 answers
206 views

Could there be a better classification of finite simple groups?

The current classification of finite simple groups puts every finite simiple group in one of a few categories. There are the "nicely" behaved infinite categories (cyclic, alternating, Lie-...
1 vote
0 answers
89 views

Irreducibility of adjoint representation

Let $ \mathbb{F} $ be a finite field of characteristic $ p\geq 5 $, $ G $ a finite group and $ \rho:G\to {\rm GL}_{2}(\mathbb{F}) $ be a representation of $ G $. By $ \text{ad}^{0}(\rho) $ we denote ...
5 votes
0 answers
156 views

Finite groups with number of generators strictly less than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $\langle a \mid a^n\rangle$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is ...
3 votes
1 answer
119 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 ...
1 vote
0 answers
127 views

Finite simple groups of order $p+1$

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Cross-post from MSE. There are some very interesting comments on the original post if you want to go check it out. Are there any well known ...
6 votes
1 answer
463 views

Finite simple groups and $ \operatorname{SU}_n $

A follow-up question to Alternating subgroups of $\mathrm{SU}_n $. $\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...

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