Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
674 questions with no upvoted or accepted answers
52
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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,...
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37
votes
0
answers
1k
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Is this generalized character always a character?
Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
- 44.4k
36
votes
0
answers
966
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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 ...
- 606
33
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0
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2k
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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-...
- 35.5k
32
votes
0
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993
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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 ...
- 3,645
28
votes
0
answers
676
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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 ...
- 22.5k
26
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0
answers
1k
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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 ...
- 156k
24
votes
0
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813
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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 ...
- 1,433
22
votes
0
answers
1k
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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 ...
- 1,241
21
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0
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473
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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.$$
This induces a linear ...
- 1,949
21
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0
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578
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Density of first-order definable sets in a directed union of finite groups
This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
- 2,200
19
votes
0
answers
604
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 $\psi:G\times G \...
user6671
18
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0
answers
2k
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$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 $|G|=2^p-1$, ...
- 4,074
18
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0
answers
524
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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 ...
- 52.9k
17
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0
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692
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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 ...
- 24.9k
17
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0
answers
824
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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, ...
- 11.2k
17
votes
0
answers
512
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 ...
- 37.7k
16
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0
answers
1k
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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 ...
- 6,290
15
votes
0
answers
189
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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 ...
- 25.5k
15
votes
0
answers
885
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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 (...
- 52.9k
14
votes
0
answers
527
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 ...
- 4,081
14
votes
0
answers
341
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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 ...
- 141
14
votes
0
answers
262
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|\...
- 2,306
14
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0
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716
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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-...
user21230
14
votes
0
answers
347
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 ...
- 54.6k
14
votes
0
answers
378
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 ...
- 50.8k
13
votes
0
answers
247
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 $n \geq 5$....
- 26.5k
13
votes
0
answers
740
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" ...
- 24.9k
13
votes
0
answers
586
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 $G \rightarrow K \leftarrow H$...
- 1,974
13
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0
answers
1k
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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 ...
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12
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0
answers
340
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
12
votes
0
answers
420
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
0
answers
191
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
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 ...
- 19.6k
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 ...
- 44.4k
12
votes
0
answers
775
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, ...
- 3,291
12
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0
answers
2k
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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 ...
- 1,196
11
votes
0
answers
722
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, ...
user6671
11
votes
0
answers
541
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 ...
- 41.8k
11
votes
0
answers
340
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 ...
- 1,838
11
votes
0
answers
400
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 ...
- 19.2k
10
votes
0
answers
424
views
Function related to length of group presentations: is it computable?
(This question comes from a friend who works in sofic group theory.)
Consider the function $f: \mathbb{N} \to \mathbb{N}$, defined, for any $n \in \mathbb{N}$, by putting $f(n)$ to be the largest ...
- 339
10
votes
0
answers
194
views
Singularity category of a hypersurface associated to $M_{11}$
For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
- 16.2k
10
votes
0
answers
134
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 ...
- 16.2k
10
votes
0
answers
368
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 ...
- 7,722
10
votes
0
answers
194
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 $n$ ...
- 7,545
9
votes
0
answers
292
views
Tilings in finite (not necessarily Abelian) groups
Let $G$ be a finite (not necessarily abelian) group. We call $A \subseteq G$ a right-tiling (for simplicity, a tiling) of $G$ if there exists a $B \subseteq G$ so that
$$ G = \bigsqcup_{b\in B} bA.$$
...
- 1,354
9
votes
0
answers
189
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 ...
- 756
9
votes
0
answers
262
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 ...
9
votes
0
answers
445
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 ...
- 44.4k