Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
Frieder Ladisch's user avatar
36 votes
0 answers
923 views

Are there infinite versions of sporadic groups?

The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group. For each of the groups of Lie ...
Myself's user avatar
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32 votes
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2k views

Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson odd-...
David Roberts's user avatar
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32 votes
0 answers
958 views

Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
DavidLHarden's user avatar
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28 votes
0 answers
657 views

Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
Peter Mueller's user avatar
25 votes
0 answers
959 views

Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$? I have no real ...
David E Speyer's user avatar
24 votes
0 answers
770 views

Revising the proof of CFSG

This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the ...
semisimpleton's user avatar
22 votes
0 answers
1k views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
William DeMeo's user avatar
21 votes
0 answers
458 views

Is there a "direct" proof of the Galois symmetry on centre of group algebra?

Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$ This induces a linear ...
Chris H's user avatar
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21 votes
0 answers
573 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
Gene S. Kopp's user avatar
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19 votes
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600 views

How is this group theoretic construct called?

Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be $$\psi(g,h) = |g|+|h|-|gh|$$ Then $\psi:G\times G \...
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18 votes
0 answers
2k views

$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?

During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you : $G$ a group, with $p$ a prime number, and $|G|=2^p-1$, ...
Dattier's user avatar
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18 votes
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509 views

Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...
Jim Humphreys's user avatar
17 votes
0 answers
678 views

Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
Alexander Chervov's user avatar
17 votes
0 answers
812 views

What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11, London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
Nick Gill's user avatar
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17 votes
0 answers
498 views

Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction. Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
Brendan McKay's user avatar
16 votes
0 answers
1k views

How many sporadic simple groups are there, really?

I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...
Simon Rose's user avatar
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15 votes
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188 views

Quantitative form of Wielandt's theorem

The following theorem was proved in [Helmut Wielandt. Eine Verallgemeinerung der invarianten Untergruppen. Mathematische Zeitschrift 45 (1939): 209-244.] a long time ago: Theorem: (Wielandt) There ...
Andreas Thom's user avatar
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15 votes
0 answers
880 views

How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question 34387 about computing the orders of finite unitary groups and the comments made there. Between 1955 (Chevalley's Tohoku paper) and 1968 (...
Jim Humphreys's user avatar
14 votes
0 answers
469 views

Is the monster group maximal in SO(196883)?

$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
Ian Gershon Teixeira's user avatar
14 votes
0 answers
322 views

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 ...
kyleormsby's user avatar
14 votes
0 answers
702 views

Algebra for the Baby

I am reading the following article. Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012.. Author works with 4370-...
user avatar
14 votes
0 answers
346 views

What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?

Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
Theo Johnson-Freyd's user avatar
13 votes
0 answers
243 views

Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
Mare's user avatar
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13 votes
0 answers
717 views

Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over "field with one element"

Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group there is well-known "natural bijection" ...
Alexander Chervov's user avatar
13 votes
0 answers
361 views

A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
Richard Stanley's user avatar
13 votes
0 answers
584 views

Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ satisfying the following conditions: $G$ and $H$ are finite groups and $K$ is an infinite group. there exist two monomorphisms $G \rightarrow K \leftarrow H$...
Ilias A.'s user avatar
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13 votes
0 answers
1k views

Cyclic Sylow $p$-subgroup in finite simple groups

I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt. All finite simple groups have a cyclic Sylow $p$-subgroup for some $p$ I found ...
Portland's user avatar
  • 2,752
12 votes
0 answers
405 views

Non-isomorphic groups with same character tables and different Brauer character tables

Let $A$ be a discrete valuation ring with perfect residue field $k$ of characteristic $p$ and field of fractions $K$ of characteristic $0$. Let $G$ and $H$ be two finite groups and assume that $K$ is ...
Sebastian A. Spindler's user avatar
12 votes
0 answers
189 views

Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following: It ...
Sebastien Palcoux's user avatar
12 votes
0 answers
699 views

Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups ...
Stefan Kohl's user avatar
  • 19.5k
12 votes
0 answers
404 views

Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...
Geoff Robinson's user avatar
12 votes
0 answers
721 views

$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
verret's user avatar
  • 3,151
12 votes
0 answers
1k views

Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$ Let $A$ be a finite abelian group and $G$ a finite group acting on $A$. Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
Damien Robert's user avatar
11 votes
0 answers
248 views

Which irreducible representations of the symmetric group are eigenspaces of class sums?

In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
Hjalmar Rosengren's user avatar
11 votes
0 answers
718 views

What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ My question is, ...
user avatar
11 votes
0 answers
520 views

Cyclic and prime factorizations of finite groups

A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$. In Cryptology factorizations of groups are known as ...
Taras Banakh's user avatar
  • 40.7k
11 votes
0 answers
321 views

$K$-theory spectrum of the category of finite groups

(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$ \newcommand{\FinGrp}{\mathbf{FinGrp}} $ Way back in my first group theory ...
Yuri Sulyma's user avatar
  • 1,513
11 votes
0 answers
396 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
JSE's user avatar
  • 19.1k
10 votes
0 answers
325 views

Is every finite group the automorphism group of a smooth projective curve?

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite group and let $k$ be a field with algebraic closure $K$. Is there a smooth projective curve $C$ defined over $k$ such that $\Aut_k(C)=\Aut_K(C)$ is ...
Jérémy Blanc's user avatar
9 votes
0 answers
177 views

Cyclic numbers of the form $2^n + 1$

A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
Z. A. K.'s user avatar
  • 586
9 votes
0 answers
114 views

Is there a strictly coassociative resolution of polynomial growth, for a finite group?

Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
Dave Benson's user avatar
  • 11.6k
9 votes
0 answers
247 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 ...
Sudarshan Narasimhan's user avatar
9 votes
0 answers
429 views

Which finite solvable groups have solvable automorphism groups?

Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...
Geoff Robinson's user avatar
9 votes
0 answers
634 views

Sufficient condition for complementation of abelian normal subgroup

Suppose that we are given a finite $p$-group $G$ and an abelian normal subgroup $A$ of $G$. The question I have is whether any sufficient conditions are known for $A$ to have a complement in $G$. From ...
the_fox's user avatar
  • 347
9 votes
0 answers
109 views

Smith normal form of conjugacy class actions

This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group. Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the symmetric group $S_n$. Identify a ...
Richard Stanley's user avatar
9 votes
0 answers
289 views

Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups

A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
Sebastien Palcoux's user avatar
9 votes
0 answers
186 views

Permutation groups with diameter $O(n \log n)$

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture: Suppose that 1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
Dmytro Taranovsky's user avatar
9 votes
0 answers
165 views

When is the rank 2 free metabelian group of exponent $n$ center free?

Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free? The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
stupid_question_bot's user avatar
9 votes
0 answers
468 views

"A remarkable Moufang loop"

The 1985 paper A simple construction of the Fischer-Griess monster group by Conway refers to an "in press" article, A remarkable Moufang loop, with an application to the Fischer group $Fi_{24}$, by ...
David Roberts's user avatar
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