Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

507 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
48
votes
0answers
878 views

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,...
35
votes
0answers
764 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 ...
30
votes
0answers
872 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 ...
29
votes
0answers
1k 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-...
28
votes
0answers
578 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 ...
25
votes
0answers
847 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 ...
22
votes
0answers
958 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 ...
21
votes
0answers
539 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 \...
18
votes
0answers
567 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 \...
18
votes
0answers
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 $|G|=2^p-1$, ...
18
votes
0answers
471 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 ...
17
votes
0answers
609 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 ...
17
votes
0answers
767 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, ...
17
votes
0answers
460 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 ...
17
votes
0answers
1k views

What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...
15
votes
0answers
172 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 ...
15
votes
0answers
483 views

Is every normal subgroup of $SL_2(\mathbb{Z}/n)$ also normal inside $GL_2(\mathbb{Z}/n)$?

Is every normal subgroup of $SL_2(\mathbb{Z}/n)$ also normal inside $GL_2(\mathbb{Z}/n)$? Of course, it suffices to ask the question when $n = p^m$ a prime power. In Classification of the Normal ...
15
votes
0answers
961 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 ...
15
votes
0answers
848 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 (...
14
votes
0answers
662 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-...
14
votes
0answers
327 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 ...
13
votes
0answers
229 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 $n \geq 5$....
13
votes
0answers
350 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 ...
13
votes
0answers
551 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 $G \rightarrow K \leftarrow H$...
12
votes
0answers
324 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 ...
12
votes
0answers
687 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, ...
12
votes
0answers
185 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 ...
12
votes
0answers
574 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" ...
12
votes
0answers
390 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 ...
12
votes
0answers
663 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, ...
12
votes
0answers
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 ...
12
votes
0answers
973 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 ...
11
votes
0answers
399 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 ...
11
votes
0answers
301 views

Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
11
votes
0answers
378 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 ...
11
votes
0answers
653 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 ...
10
votes
0answers
188 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|\...
10
votes
0answers
238 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 ...
9
votes
0answers
322 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 ...
9
votes
0answers
506 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 ...
9
votes
0answers
99 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 ...
9
votes
0answers
133 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 $n$ ...
9
votes
0answers
158 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 $(\...
9
votes
0answers
400 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 ...
9
votes
0answers
277 views

An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
8
votes
0answers
99 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 ...
8
votes
0answers
243 views

Does there exist a nontrivial perfect group with a "locally commuting" presentation?

EDIT: I originally insisted that the perfect group in question be finite, however I now realize that I do not need this condition, only that the generators used in the presentation have finite order. ...
8
votes
0answers
198 views

Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(\mathbb{F}_{p})$

Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear group and $U_{n}$ denote the unitriangular group of $n\times ...
8
votes
0answers
132 views

Finite groups with unbounded commutator width and no nontrivial central chief factor

The commutator width of a group is the smallest $n$ such that every product of commutators is a product of $n$ commutators. My initial question was: Do there exist finite perfect groups with ...
8
votes
0answers
245 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'...

1
2 3 4 5
11