# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,933
questions

**4**

votes

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### Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...

**18**

votes

**2**answers

1k views

### Atypical use of Sylow?

The typical application of Sylow's Theorem is to count subgroups. This makes it difficult to search the web for other applications, since most hits are in the context of qualifying exams.
What are ...

**3**

votes

**1**answer

149 views

### Finite maximal closed subgroups of Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SO{SO}$
Let $G$ be a Lie group.
I am interested in maximal closed subgroups $ G $ which happen to be finite.
The ...

**4**

votes

**1**answer

257 views

### Group presentation in the category of finite group

Context: I'm trying to deal with presentations in the framework of Gonthier et al. formalization of the group theory in the proof assistant Coq. It was used to machine check the Feit-Thompson odd ...

**1**

vote

**0**answers

58 views

### Tower of $p$-groups

The number of isomorphism classes of groups of order $p^n$ grows so fast $\big (p^{{\frac{2}{27}}n^{3}+O(n^{8/3})} \big)$,
that a folklore conjecture asserts that, asymptotically, almost every finite ...

**8**

votes

**1**answer

295 views

### Chapter 4 Section 2 of Macdonald's Symmetric Functions and Hall Polynomials

Throughout this post $G$ denotes $GL_{n}(\mathbb{F})$ where $\mathbb{F}$ denotes the finite field of $q$ elements.
I'm currently reading the aforementioned book to understand how the irreducible ...

**1**

vote

**0**answers

31 views

### Group graphs and Ramsey Theory. Sub-question 2

This note is a continuation of Group graphs and Ramsey theory. Sub-question 1.
Let $\ X\ $ be a group, and let $\ c:\binom X2\to C\ $ be a two-coloring ($r\ $ and $\ g\ $ are the two colors). ...

**0**

votes

**1**answer

152 views

### Twisted forms of $\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...

**11**

votes

**2**answers

828 views

### Low-order symmetric group 2-generation: n=5,6,8

In a comment at the recent question What is the standard 2-generating set of the symmetric group good for?, it was remarked that the symmetric groups $S_n$ for $n\gt 2$, $n\neq 5,6,8$, can be ...

**22**

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**6**answers

2k views

### What is the standard 2-generating set of the symmetric group good for?

I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...

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116 views

### General linear group analogs

The Wikipedia pages for $E_6$ and $E_7$ list three series of groups notated as each of $E_6(q)$, $^2E_6(q)$, and $E_7(q)$:
The simple form, analogous to $\operatorname{PSL}_n(q)$
The adjoint form, ...

**1**

vote

**2**answers

426 views

### What are all the transitive extensions of cyclic groups?

"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the ...

**2**

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**0**answers

103 views

### General lower bound on the number of subgroups of a finite group

The general question is this: given a positive integer $n$, are there any non-trivial lower bounds on the number of subgroups of a group of order $n$?
Some more specific thinking: we know that in the ...

**39**

votes

**1**answer

1k views

### Known and fixed gaps in the proof of the CFSG

As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this ...

**0**

votes

**1**answer

120 views

### Computationally intractable orbit of a monoid action on a finite set

Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...

**2**

votes

**2**answers

106 views

### Smallest $\mathbb R$-algebra which contains a subgroup isomorphic to $A_4$

$A_4$ (the alternating group on $4$ elements) can be thought of as the group of direct Euclidean isometries of a regular tetrahedron. This shows that there is a subgroup of the algebra of $3\times3$ ...

**2**

votes

**1**answer

114 views

### Automorphism groups of simple groups of Lie type

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a ...

**1**

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**0**answers

93 views

### Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?

In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group.
Suppose: $[l,r]:x\to \bar lxr\;,\...

**49**

votes

**2**answers

5k views

### Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?

According the the Wikipedia page, the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher ...

**2**

votes

**0**answers

133 views

### Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...

**6**

votes

**1**answer

270 views

### On the density of the orders excluded by the Sylow theorems for simple groups

If $G$ is a finite group whose order is divisible by a prime $p$ and $p^r$ is the maximal power of $p$ that divides it, the Sylow theorems tell us that the number $n_p$ of Sylow $p$-subgroups of $G$ ...

**4**

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136 views

### Subgroups of $\operatorname{GL}(n,q)$ transitive on non-zero vectors

Is there a classification of subgroups $G$ of $\operatorname{GL}(n,q)$ which act transitively on $\mathbb{F}_q^n \setminus \{0\}$, the set of non-zero vectors?
Any $G$ with $\operatorname{GL}(n/m,q^m) ...

**7**

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178 views

### Classification of octonionic reflection groups

I know that there exist classification theorems for real, complex, and quaternionic, reflection groups.
There are presentations for the real reflection groups, as well as further presentations for the ...

**0**

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**0**answers

40 views

### How can I find the order of the elements of the maximal subgroups for G_2(3)?

I'm looking to find the maximal subgroups for the exceptional group of Lie type $G_{2}(3)$ using GAP.
Currently I can do the following:
...

**0**

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**0**answers

36 views

### Counting the number of generating triples of various types in finite simple groups

I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ...

**7**

votes

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96 views

### What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?

Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...

**8**

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180 views

### Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?

Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...

**3**

votes

**1**answer

69 views

### Bounds for the number of edges in an Alperin diagram

If $A$ is an algebra over a field $k$ and $M$ is a finite-dimensional $A$-module, then Alperin showed in a paper [Diagrams for modules, JPAA, 1980] how to associate a diagram to $M$ with the vertices ...

**2**

votes

**1**answer

114 views

### How do I find hyperbolic generating triples for a group using GAP?

Let $G$ be a finite group and $x, y, z \in G$. A hyperbolic generating triple for $G$ is a triple $(x, y, z) \in G\times G\times G$ such that
$\frac{1}{o(x)}+\frac{1}{o(y)}+\frac{1}{o(z)} <1$,
$\...

**4**

votes

**3**answers

426 views

### What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:
Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...

**25**

votes

**4**answers

993 views

### $\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in $\mathfrak{S}_p$ for $p>11$

A famous result of Galois, in his letter to Auguste Chevalier, is that for $p$ prime $>11$ the group $\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in the symmetric group $\mathfrak{S}_p$. ...

**17**

votes

**3**answers

3k views

### Finite groups such that every irrep can be induced from trivial irrep of a subgroup ?

What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from trivial ...

**41**

votes

**8**answers

9k views

### The finite subgroups of SU(n)

This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ...

**15**

votes

**2**answers

575 views

### The finite groups with a zero entry in each column of its character table (except the first one)

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...

**9**

votes

**0**answers

109 views

### Tensor products of irreducible representations of $GL_{2}(\mathbb{F}_{q})$

Throughout the post $G = GL_{2}(\mathbb{F}_{q})$ where $q$ is a prime power with the prime not being 2.
Let $V_{1}$ and $V_{2}$ be cuspidal representations of $G$ over $\mathbb{C}$. I can understand ...

**6**

votes

**0**answers

139 views

### Subalgebra of group algebra generated by idempotents

Let $G$ be a finite group, and let $A$ and $B$ be two abelian subgroups of $G$. Let $K$ be a number field such that all characters of $A$ and of $B$ take values in $K$. Let $\mathcal{O}_K$ be the ring ...

**9**

votes

**1**answer

238 views

### How small can the support of a nontrivial $\mathbb F_p$-cocycle on $C_p$ be?

Let $p$ be a prime, and let $\phi : C_p^n \to \mathbb F_p$ be an $\mathbb F_p$-valued $n$-cocycle on $C_p$ (the cyclic group of order $p$) which is not an $n$-coboundary, i.e. $\phi$ represents a ...

**30**

votes

**3**answers

3k views

### 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 $...

**12**

votes

**3**answers

430 views

### Small simplicial set models for BG

Let $F$ be a finite group.
Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?
For example the Bar construction has the ...

**2**

votes

**0**answers

63 views

### The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$

I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...

**8**

votes

**2**answers

1k views

### Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...

**1**

vote

**2**answers

337 views

### Why are finite simple groups useful? [duplicate]

The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of ...

**25**

votes

**1**answer

2k views

### Number of 2-dimensional irreducible representations of a finite group ?

Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ).
...

**0**

votes

**1**answer

151 views

### Irreducible representations of finite p-groups

Let $G$ be a finite $p$-group. What are irreducible representations of $G$ over a field of characteristic $q$, such that $(p,q)=1$ ? Can we say something in general ? In particular, if there exists ...

**3**

votes

**3**answers

445 views

### Large product-1-free sets in finite groups

$\DeclareMathOperator\SmallGroup{SmallGroup}$Definition. A subset $A$ of a group $G$ is called product-1-free if for any sequence of pairwise distinct elements $a_1,\dots,a_n$ of $A$ the product $a_1\...

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**0**answers

108 views

### Another question concerning finite metacyclic groups

Given a non-split finite metacyclic group $H$, does there always exist a finite split metacyclic group $G$ with a normal cyclic subgroup $N$ of prime power order such that $H \cong G/N$?
Based on my ...

**2**

votes

**1**answer

158 views

### Structures of subgroups of a finite abelian p-group

$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...

**0**

votes

**1**answer

92 views

### 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 $n$. ...

**4**

votes

**1**answer

178 views

### The action of a subgroup of the torsion group of elliptic curves on integral points?

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a ...

**4**

votes

**1**answer

203 views

### Finite simple groups with the same numbers of elements of orders p and q

Let $G$ be a nonabelian finite simple group, and let $p$ and $q$ be
distinct prime divisors of the order of $G$. Is it true that the
number of elements of $G$ of order $p$ never equals the number of ...