# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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3 votes
1 answer
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5 votes
1 answer
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### 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

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

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

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

4 votes
4 answers
3k views

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, ...