# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

2,057
questions

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### 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
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### 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
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### 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 ...

6
votes

1
answer

233
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### "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 ...

16
votes

2
answers

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### Groups with all normal subgroups characteristic

Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me ...

5
votes

0
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75
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### 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 ...

11
votes

3
answers

662
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### Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$?
Same question, but this time $G$ is a finite group with at most $c$...

2
votes

1
answer

122
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### 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?

4
votes

0
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185
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### 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 ...

3
votes

0
answers

244
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### 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 \...

0
votes

0
answers

181
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### 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-...

3
votes

1
answer

662
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### Unique factorization of finite groups under direct sum?

I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then ...

-1
votes

2
answers

214
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### 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 ...

11
votes

1
answer

516
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### Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the ...

6
votes

1
answer

236
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### Presentation of the Monster as a Hurwitz group

The Monster group is the largest of the sporadic simple groups, and has been proven by Wilson to also be a Hurwitz group. It has a presentation in terms of Coxeter groups, specifically Y443 along with ...

3
votes

1
answer

232
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### What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?

An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...

60
votes

5
answers

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### Heuristic argument that finite simple groups _ought_ to be "classifiable"?

Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down?
Solomon's AMS article goes some way toward a historical / technical ...

2
votes

2
answers

170
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### 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 ...

5
votes

1
answer

329
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### 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,...

18
votes

1
answer

1k
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### 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 ...

4
votes

1
answer

165
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### 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 ...

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 $x\in X$. ...

2
votes

0
answers

44
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### 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 ...

31
votes

3
answers

3k
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### Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order $...

0
votes

0
answers

35
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### 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

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 ...

3
votes

0
answers

112
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### $2^2 \cdot U_6(2)$ 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
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### 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 ...

4
votes

0
answers

55
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### 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
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### 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 $...

14
votes

2
answers

701
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### 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 ...

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 ...

3
votes

1
answer

175
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### Boolean ring of unitary divisors / Structure of unitary divisors?

I hope this question is appropriate for MO:
Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors.
We can make $U_n$ to a boolean ring:
$$a \...

3
votes

0
answers

117
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### 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
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### 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
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### 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
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### 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

129
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### 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
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### 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
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### 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 ...

89
votes

14
answers

31k
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### Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

6
votes

0
answers

112
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### 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
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### 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

76
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### 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\...

3
votes

2
answers

524
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### Complexity of establishing finite groups (non)-isomorphism ?

Question Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N ...

6
votes

3
answers

476
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### Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$

Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...

16
votes

3
answers

946
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### Reference for representation theory of SL_2(Z/n)

There are many references for the representation theory (say over $\mathbf C$) of $\operatorname{SL}_2(\mathbf{F}_q)$ and $\operatorname{GL}_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris &...

0
votes

1
answer

137
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### Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...

27
votes

4
answers

6k
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### Classification of finite groups of isometries

Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$.
For $n=2$ it is cyclic and dihedral groups.
For $n=3$ they are well known, probably from Kepler and are related ...

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(...