Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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4 votes
1 answer
215 views

How to make Burnside's formula compatible with point counting for varieties over finite fields?

If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as: $$ |X/G| = \frac1{|G|} \sum_{g\in G} |X^g|, $$ with $X^g$ being the set of ...
1 vote
0 answers
59 views

Second homology group of a presentation complex

I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly. Given a finite group $G$, and a presentation $P$ of ...
14 votes
0 answers
254 views

Is this class of groups already in the literature or specified by standard conditions?

In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I ...
5 votes
0 answers
75 views

Endo reversible words

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if ...
  • 183
6 votes
1 answer
233 views

"Novelty" maximal subgroups in $S_n$

What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$? Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...
  • 2,536
4 votes
0 answers
185 views

Finite 2-groups with $(ab)^{2}=(ba)^{2}$

There exist nonabelian finite 2-groups $G$ with the property $(A2)$ : for every $a,b\in G$, $(ab)^{2}=(ba)^{2}$. An example of a such group is given by the quaternion group $Q_{8}$ of order 8. Is ...
  • 913
0 votes
0 answers
181 views

Groups of orders $7!$ and $\frac{7!}{2}$

In our research, we need to know that whether every group $G$ of order $2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7=\frac{7!}{2}$ or $5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7=7!$ has a proper subgroup non-...
-1 votes
2 answers
213 views

Splitting of a finite group with no abelian subfactor in composition series

Let $G$ be a finite group with no abelian subfactor in its composition series. Is $G$ obtained from simple groups by iterating semidirect products? (Initially it was asked whether $G$ is a direct ...
  • 149
3 votes
0 answers
244 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 \...
  • 2,536
5 votes
1 answer
329 views

The number of polynomials on a finite group, II

This question is follow up of this MO-post. First let us recall the necessary definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0,...
  • 34.8k
2 votes
0 answers
44 views

Finite groups whose polynomials share two common properties with polynomials on commutative groups

This question is motivated by (some available information on) this MO-problem on the largest possible degree of a polynomial on a finite group and this MO-problem on the degree of the constant ...
  • 34.8k
9 votes
1 answer
440 views

The degree of a constant polynomial on a finite group

A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
  • 34.8k
4 votes
1 answer
165 views

Prime divisors of nonabelian simple group and of its outer automorphism group

Let $G$ be a finite nonabelian simple group. Write $\mathrm{Out}(G)$ the outer automorphism group of $G$. For a finite group $H$, let $\pi(H)$ be the prime divisors of the order of $H$. By check the ...
  • 253
0 votes
0 answers
35 views

Polyextremal groups

A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form $f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
  • 34.8k
3 votes
1 answer
107 views

Length of representation of $GL_n(\mathbb{F}_q)$ in functions on Grassmannian

Let $G=GL_n(\mathbb{F}_q)$ be the (finite) group of all linear invertible transformations of the vector space $(\mathbb{F}_q)^n$ over the finite field $\mathbb{F}_q$. $G$ acts naturally on the ...
  • 19.3k
3 votes
0 answers
112 views

$2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$ in $\mathbb{M}$

In the first diagram of this paper, there are conjugacy classes of subgroups of the Monster group which are labeled $2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$, respectively. Can subgroups in the ...
0 votes
0 answers
48 views

Investigating the structure of a group algebra via the derived subgroup

It is well known that each element in the special linear group $\mathrm{SL}_n(\mathbb{H})$ over the real quaternion division ring with $n\geq1$ is a single multiplicative commutator. I am particularly ...
14 votes
2 answers
701 views

Which finite groups have low-degree essential cohomology?

Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
4 votes
0 answers
55 views

References for completions of finite group tensor categories

Let $G$ be a finite group and $\operatorname{Vec}_G$ be the tensor category of $G$-graded vector spaces (or, if you prefer, $\pi_{\le 2}(BG)$). The completion $\overline{\operatorname{Vec}_G}$ of $\...
1 vote
0 answers
129 views

Realization of a subgroup in a maximal subgroup of a classical group

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...
1 vote
0 answers
83 views

Number of ways to write a group element as a product of generators

Let $G$ be a finite group generated by some finite set $S = \{g_1, g_2, ...\} \subseteq G$. Let $h \in G$ be some element. Let the function $c_n: G \rightarrow \mathbb{N}$ be defined that $c_n(h)$ is ...
  • 111
3 votes
0 answers
117 views

Quotient of $\mathbb P^n$ by the symmetric group $S_{n+1}$

The projective space ${\mathbb P}^n$ of dimension $n$ over a field (let's take $\mathbb C$ for simplicity) can be viewed as the space of homogeneous coordinates $[x_0:\cdots :x_n]$ in the $n+1$ ...
4 votes
0 answers
132 views

Large subsets of groups with no solution to linear equations

Is there a (sequence of finite nonabelian) group(s) $G$ and a (sequence of corresponding) subset(s) $S \subseteq G$, $|S| = |G|^{1-o(1)}$, such that there is no solution to $xy^{-1}z = zy^{-1}x$ with ...
11 votes
2 answers
660 views

Finite groups with integral character table

The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
5 votes
1 answer
159 views

Is the derived group of the G(F) perfect

Let $G$ be a connected reductive group defined over a finite field $F$ of characteristic $> 3$. Is it true that the commutator group of $G(F)$ is perfect? This is true if $G$ is assumed to be ...
  • 2,277
6 votes
1 answer
129 views

Is the largest normal abelian subgroup of a finite 2-group $P$ of order at least the square root of the order of $P$?

Let $G$ be a group of order $2^n$. Does $G$ have a normal abelian subgroup of order at least $2^{n/2}$? (This is true, via computations in GAP, for $n \le 8$. The question is similar to one posed ...
2 votes
0 answers
214 views

Existence of $\sqrt{2}$ in a finite group algebra over $\mathbb{Q}$

I cannot find a finite group $G$ such that $\exists x\in \mathbb{Q}[G]$ with $x^2=2e$, where $\mathbb{Q}[G]$ is the group algebra of $G$ over $\mathbb{Q}$. I also could not prove it does not exist. ...
5 votes
1 answer
348 views

Is any finite-dimensional algebra a sub-algebra of a finite-group algebra?

For $A$ a finite-dimensional algebra over a field $K$ Does there exist a finite group $G$, such that $A$ is a sub-algebra of $K[G]$ ? Where $K[G]$ denotes the group-algebra of $G$ over $K$. In case ...
6 votes
0 answers
112 views

When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (...
2 votes
0 answers
78 views

G graph connections for finite groups G

In my research, I have seen G graph connections usually when G is a Lie group and the graph is the fatgraph of a (punctured) surface. This is usually in a physics context. However, I am curious to ...
  • 129
1 vote
0 answers
76 views

Minimizing distance over finite group action

Let $G$ be a finite group and $V$ a unitary irreducible rep’n of dimension $N$. Is there a fast (polynomial in $\log|G|$) algorithm to compute $\displaystyle \min_{g \in G}d(x,gy)=\max_{g \in G} Re\...
2 votes
0 answers
60 views

Positive values of Schur polynomials

Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials $$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...
6 votes
2 answers
232 views

Differences between $p$-groups and $q$-groups

First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite. Academically, I work with connecting the arithmetic structure of ...
0 votes
0 answers
77 views

What are the maximal closed subgroups of $ SU_3 $?

What are the maximal closed subgroups of $ SU_3 $? This question is inspired by Lie subgroups of SU(3). Interesting partial answers to that question, treating only the case of connected subgroups, are ...
3 votes
1 answer
119 views

Inclusions among finite orthogonal groups over finite fields

I am looking for a reference. I hope that what follows is in some textbook. Let $q$ be an odd prime power and let $\ell$ be a positive integer. Now, let $\mathfrak{q}:\mathbb{F}_{q^\ell}^2\to\mathbb{F}...
0 votes
0 answers
41 views

$p$-centric subgroups control fusion

Given two finite groups $G,H$ such that $$\displaystyle\bigoplus_{Q\cong L\leq G\ \mathrm{up\ to}\ G\mathrm{-conjugation}}{\mathbb{F}_p\Bigg[\frac{\mathrm{Out}(L)}{\mathrm{Out}_G(L)}\Bigg]}\cong\...
  • 83
2 votes
1 answer
194 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(...
  • 2,171
8 votes
1 answer
353 views

Linear recurrence relation for symmetric powers in the Burnside ring

Let $G$ be a finite group and $B(G)$ be its Burnside ring, i.e. formal sums of isomorphism classes of finite $G$-sets with addition given by disjoint union and multiplication given by Cartesian ...
1 vote
1 answer
189 views

action of the extra-special group

I'm reading a paper which has this line: A direct computation shows that $P\Omega_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C_{P\Omega_8(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
3 votes
1 answer
127 views

The torsion subgroup of the coinvariants for a $G$-module

Let $G$ be a finite group and $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. Consider the functor $$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
2 votes
1 answer
122 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?
7 votes
0 answers
158 views

Finite group with exactly one class each of two given groups

Given two nonisomorphic finite groups, is it always possible to construct a larger finite group with exactly one conjugacy class of subgroups isomorphic to each?
4 votes
1 answer
169 views

Irreducible deleted permutation module for a finite group

Let $G$ be a subgroup of the symmetric group $S_n = \operatorname{Sym}(X)$. Let $k$ be a field, and let $V$ be the permutation module corresponding to $X$. Then $V$ is not irreducible, it has a $1$-...
  • 2,536
3 votes
3 answers
184 views

Perfect group that is split extension of a normal free subgroup of finite index

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect? Thanks @YCor for reformulating the question.
  • 555
2 votes
0 answers
35 views

Brauer pairs associated to a normalizer subsystem in the fusion system of a block of a finite group

Let $G$ be a finite group and let $k$ be an algebraically closed field of positive characteristic $p$. Let $b$ be a block of $kG$ and let $(P,e)$ be a maximal $(G,b)$-Brauer pair. For every subgroup $...
5 votes
1 answer
177 views

What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
-4 votes
1 answer
127 views

Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$

How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?
2 votes
2 answers
170 views

Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?

The background: We recall/define the following: $\Omega_n=\{1,\dots,n\}$. $M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups ...
  • 1,008
0 votes
1 answer
185 views

The center of Sylow subgroups

$\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. And $G/\Phi(G)$ is a simple group. Let $P\in \Syl_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and $\...
  • 21
4 votes
0 answers
91 views

Is there a cohomological interpretation of the bilinear form arising from Clifford theory?

For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
  • 1,557

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