# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\operatorname{PSL}(2,\mathbb{F}_p)$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $p$...
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 ... ?...