Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,259
questions
2
votes
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128
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The property of subgroups of a finite solvable group
$\DeclareMathOperator\Syl{Syl}$$G$ is a finite solvable group and $G=PQR$, where $P\in \Syl_{p}(G)$, $Q\in \Syl_{q}(G)$, $R\in \Syl_{2}(G)$ and $|R|=2$. Suppose that $C_P(R)=P$ and $C_Q(R)=1$.
Since $...
2
votes
0
answers
363
views
Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
1
vote
0
answers
62
views
Distinguish $p'$-elements in a coset
Let $G$ be a finite group with a nonabelian minimal normal subgroup $N$.
Then $N$ is a direct product of $n$ copies of some nonabelian simple group $S$.
Let $p$ be a prime divisor of $|S|$ and let $xN\...
8
votes
1
answer
439
views
I M Isaacs Algebra Exercise 9.4
I am a PhD student in the represention theory of finite groups. One of my friends and I solved all exercises in the book I M Isaacs - Algebra A Graduate Course except for the following exercise in ...
3
votes
1
answer
327
views
Find an example where a subset of “inverse fixed points“ is not a subgroup
$G$ is a group of odd order, $\sigma$ is an automorphism of $G$, and $\sigma^2=\mathrm{id}$.
I want to find an example to show that
$G_s= \{ g \in G \mid \sigma \left( g \right)= g^{-1} \} $ might not ...
2
votes
0
answers
94
views
Number of prime factors of the order of an increasing sequence of finite non-abelian simple group
I recently come across this question: Number of prime factors of the order of a finite non-abelian simple group. What caught my attention is the second question:
Does there exist a sequence $\{S_n\}$
...
2
votes
1
answer
112
views
If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$
Let $n=2m$. What is the order of the following permutation $\sigma$?
$$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$$ to be the maximum value over the indices $[H_{i+1} : H_i]$ ...
5
votes
0
answers
134
views
The orders of which nonabelian finite simple groups can be written as products of other such orders?
Is it true that the order of a nonabelian finite simple group $G$ can be
written as the product of the orders of two or more other nonabelian
finite simple groups if and only if
$G$ is either an ...
1
vote
0
answers
113
views
Is the commutator of the holomorph of generalized quaternion group abelian?
Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
3
votes
0
answers
248
views
Commuting real elements in finite groups
Let $p$, $q$, $r$ be three distinct odd primes, and $G$ a finite group with $|G|$ divisible by $p$, $q$, $r$ to the first power only. Let $x,y,z \in G$ be of order $p,q,r$ respectively. Assume
(a.) $[...
4
votes
0
answers
129
views
Derived subgroup of rational points vs. rational points of derived subgroups
Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion
$$
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})$?
8
votes
0
answers
211
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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 ...
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?
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 "...
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_{...
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
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_{...
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 ...
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.
(...
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 ...
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. ...
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). ...
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
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^{...
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 ...
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
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 ...
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 ...
-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$ ...
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$-...
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 ...
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 ...
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}$)?
3
votes
0
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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 $...
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}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
2
votes
0
answers
128
views
Homogeneity of finite simple groups
According to A complete classification of finite homogeneous groups, the finite simple groups $G$ which are homogeneous (meaning every isomorphism between subgroups extends to an automorphism of $G$) ...
13
votes
1
answer
416
views
Locally finite groups containing all finite groups
Say that a group is rich if it contains isomorphic copies of all finite groups.
It is easy to produce rich groups, and also rich locally finite groups, for instance the restricted direct product $A=\...
2
votes
1
answer
220
views
Invariants of the group algebra of a finite group
Consider a finite group $G$ and its complex group algebra $V_G$, on which $G$ acts. I would like to know: what are the polynomial $G$-invariants of $V_G$ i.e., the polynomial functions $p\in \mathbb{C}...
0
votes
1
answer
126
views
Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
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 ...
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 ...
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 ...
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} $...
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 ...
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 ...
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-...