Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,259
questions
5
votes
0
answers
184
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 ...
0
votes
0
answers
95
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:...
1
vote
0
answers
138
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 ...
3
votes
1
answer
227
views
Extensions of a simple group by an elementary abelian $p$-group
Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other ...
5
votes
1
answer
281
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
144
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
322
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 ...
7
votes
0
answers
113
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 ...
8
votes
1
answer
295
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 ...
6
votes
1
answer
367
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 ...
0
votes
0
answers
196
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
310
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 ...
4
votes
1
answer
325
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 \...
5
votes
1
answer
355
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,...
2
votes
0
answers
46
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 ...
9
votes
1
answer
475
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 $x\in X$. ...
4
votes
1
answer
211
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 ...
0
votes
0
answers
41
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 ...
3
votes
1
answer
147
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 ...
3
votes
0
answers
123
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 ...
13
votes
2
answers
775
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 ...
5
votes
0
answers
77
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
135
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
113
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 ...
4
votes
0
answers
218
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
148
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 ...
12
votes
2
answers
840
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
193
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 ...
6
votes
1
answer
162
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
239
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
425
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 ...
7
votes
0
answers
136
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
83
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 ...
1
vote
0
answers
93
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
113
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
302
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 ...
4
votes
1
answer
535
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 ...
3
votes
1
answer
140
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}...
5
votes
1
answer
330
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(...
9
votes
1
answer
394
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
197
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
262
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
154
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
168
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?
5
votes
1
answer
220
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 $1$-...
3
votes
3
answers
245
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.
2
votes
0
answers
50
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
210
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
138
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
229
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 ...