Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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4
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0answers
83 views

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
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2answers
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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0answers
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
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1answer
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
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2answers
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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6answers
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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0answers
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
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2answers
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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0answers
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
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1answer
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 ...
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1answer
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
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2answers
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
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1answer
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 ...
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0answers
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
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2answers
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
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0answers
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
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1answer
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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0answers
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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0answers
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 ...
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0answers
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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0answers
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
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0answers
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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0answers
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
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1answer
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
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1answer
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
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3answers
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
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4answers
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
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3answers
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
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8answers
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
2answers
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
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0answers
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
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0answers
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
1answer
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
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3answers
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
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3answers
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
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0answers
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
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2answers
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
2answers
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
1answer
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
1answer
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
3answers
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\...
0
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0answers
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
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1answer
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
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1answer
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
1answer
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
1answer
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 ...

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