Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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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 $...
Moomoo Angel line's user avatar
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 ...
Dr. Evil's user avatar
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1 vote
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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\...
user44312's user avatar
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8 votes
1 answer
439 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 ...
Shi Chen's user avatar
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3 votes
1 answer
327 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 ...
zhjzwlys's user avatar
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 $\{S_n\}$ ...
W4cc0's user avatar
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2 votes
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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 $\sigma$? $$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$
ABB's user avatar
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1 vote
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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 $[H_{i+1} : H_i]$ ...
tim's user avatar
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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 ...
Stefan Kohl's user avatar
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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 $\operatorname{Hol}(Q_{2^{n+1}})$ ...
bidermeyer's user avatar
3 votes
0 answers
248 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.) $[...
Nick's user avatar
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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 ...
Dr. Evil's user avatar
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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 $\mathrm{SL}_2(\mathbb{H})$?
Daniel Sebald's user avatar
8 votes
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211 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 ...
babu_babu's user avatar
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8 votes
1 answer
449 views

Small subgroups of the monster

Is every group of order at most 36 isomorphic to a subgroup of the monster group?
Daniel Sebald's user avatar
5 votes
1 answer
178 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 "...
Nataniel Marquis's user avatar
2 votes
1 answer
110 views

Primal identity in matrix semigroup

Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product. We call $s_1\cdots s_k$ an identity index if $M_{s_1}M_{...
gondolf's user avatar
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2 votes
1 answer
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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{...
Nobody's user avatar
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190 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_{...
dm82424's user avatar
  • 350
4 votes
0 answers
186 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 ...
emiliocba's user avatar
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7 votes
1 answer
278 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. (...
Zach Hunter's user avatar
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5 votes
1 answer
274 views

Compact Lie group has finitely many Lie primitive subgroups

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper ...
Ian Gershon Teixeira's user avatar
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. ...
ReverseFlowControl's user avatar
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 ...
benblumsmith's user avatar
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2 votes
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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). ...
Colin Reid's user avatar
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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 ...
Daniel Sebald's user avatar
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}^...
Jackson Walters's user avatar
3 votes
1 answer
192 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^{...
user488802's user avatar
1 vote
1 answer
267 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 ...
Moomoo Angel line's user avatar
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^...
Daniel Sebald's user avatar
5 votes
1 answer
412 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 ...
Martin Brandenburg's user avatar
1 vote
0 answers
173 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 ...
Shi Chen's user avatar
  • 195
-4 votes
1 answer
133 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 $r$ ...
san's user avatar
  • 93
1 vote
1 answer
232 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 $2$-...
user488802's user avatar
6 votes
0 answers
312 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 ...
Sebastien Palcoux's user avatar
1 vote
0 answers
226 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 ...
gola vat's user avatar
  • 179
4 votes
1 answer
228 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 $4^{11}$)?
Daniel Sebald's user avatar
3 votes
0 answers
64 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 $...
Agelos's user avatar
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1 vote
0 answers
115 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 $1 \leqslant i \leqslant [n/2]$. ...
user488802's user avatar
2 votes
0 answers
128 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 $G$) ...
Daniel Sebald's user avatar
13 votes
1 answer
416 views

Locally finite groups containing all finite groups

Say that a group is rich if it 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=\...
YCor's user avatar
  • 60.1k
2 votes
1 answer
220 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}...
user493645's user avatar
0 votes
1 answer
126 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})^{\...
user488802's user avatar
1 vote
0 answers
59 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 ...
user488802's user avatar
4 votes
1 answer
246 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 ...
Infinity_hunter's user avatar
11 votes
2 answers
398 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 ...
Ian Gershon Teixeira's user avatar
7 votes
1 answer
533 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-1} $...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
98 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 ...
freeRmodule's user avatar
  • 1,025
2 votes
0 answers
151 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 ...
gola vat's user avatar
  • 179
1 vote
0 answers
286 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-...
Takirion's user avatar
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