# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,932
questions

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### 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 ...

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40 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 ...

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**1**answer

158 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 ...

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25 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). ...

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280 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 ...

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**1**answer

104 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 ...

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**1**answer

146 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^...

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818 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 ...

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114 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, ...

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### 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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102 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 ...

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102 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 $3\times3$ ...

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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 $...

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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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113 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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90 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\;,\...

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131 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 ...

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269 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 $G$ ...

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135 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) ...

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176 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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38 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:
...

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95 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 ...

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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 ...

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177 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;...

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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
$\frac{1}{o(x)}+\frac{1}{o(y)}+\frac{1}{o(z)} <1$,
$\...

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988 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 $\mathfrak{S}_p$. ...

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573 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 \...

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106 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 ...

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137 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 ...

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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 ...

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60 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 ...

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336 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 ...

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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 ...

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150 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 ...

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107 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 ...

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156 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 ...

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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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187 views

### A transitive action on a specific set

Let $G$ be a finite group and $\lambda\in G$, consider a set $$D^{p+*}_{G}(\lambda):=\{P\in S^{p+*}_G|P^{\lambda}=P=[P,\lambda]\}$$ where:
$S^{p+*}_{G}$ denotes the set consisting of all non-trivial $...

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444 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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113 views

### Subgroups and representations of finite groups of Lie type

Is there a usable bound for the minimal index of a proper subgroup in a finite simple group of Lie type in terms of its rank and the characteristic (or even cardinality) of its field of definition?
...

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88 views

### Action of diagonal automorphisms on the set of irreducible characters of $D_n(q)$

Let $S$ be $D_n(q)$ where $q$ is a prime power. We know that a diagonal automorphism $\phi_h$ of $S$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ normalizes $S$. Note that $\...

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117 views

### Finite simple groups of automorphisms of finite simple Lie algebras

I begin by briefly recalling some basic facts in order to pose my question in context.
According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...

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322 views

### Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$

My question is about the Hamming Weight (or number of 1's in binary expansion) of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ A152007
For example, $a_3 = 9709 = (10110111101001)_2 $ has nine 1's in binary ...

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465 views

### Product-one sets in non-commutative groups

A nonempty subset $D$ of a group $G$ is called
$\bullet$ decomposable if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$;
$\bullet$ ...

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205 views

### Finite groups with a dihedral maximal subgroup

Suppose $G$ is a finite group with a dihedral maximal subgroup. Suppose that $G$ is not isomorphic to $\operatorname{PSL}(2,q)$ for some any prime-power $q$. Is $G$ always solvable?

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151 views

### Is there always a simple module whose Green correspondent is a simple module under some conditions?

Let $G$ be a finite group and $KG$ its group algebra over some field $K$ with $\mathrm{char}\ K$ dividing the order of $G$. It's well-known that the Green correspondence is compatible with the Brauer ...

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256 views

### Is the fixed subring a symmetric algebra?

Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A,...

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68 views

### Is Broué's abelian defect conjecture true for finite groups with abelian TI Sylow p-subgroups?

I am now interested in Broué's abelian defect conjecture and I have read many papers concerning it. For a prime $p$, I informally define a finite group to be a $p$-ATI-group if it has abelian Sylow $p$...

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288 views

### When is the augmentation ideal projective as RG-module?

Let $G$ be a finite group and let $R$ be a commutative ring.
I'd like to ask, if there is a theorem of the following kind:
The augmentation ideal $I_G$ is projective as RG-module, if and only if ... ?...

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177 views

### Inverse Galois problem on simple groups

Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group.
I've tryied to mess with the embedding problem for ...