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2
votes
2answers
62 views

How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
9
votes
2answers
321 views

Certain signed sum over $S_n$

The following question appeared in my research: Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum $$ \sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1) $$ that is, we only consider ...
-3
votes
0answers
75 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$ [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
0answers
94 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
2
votes
2answers
227 views

Is anything known about the eigenspectrum of the regular representation of the permutation group?

I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known? The ...
2
votes
0answers
74 views

Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
3
votes
1answer
425 views

learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups ...
4
votes
1answer
194 views

Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer. Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
1
vote
1answer
126 views

Generating finite groups using subgroups

For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a_1, \dots, a_n \in K$ such that $\langle L, a_1 \dots, a_n \rangle = K$. Is there some $m ...
1
vote
0answers
58 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
3
votes
0answers
246 views

Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
25
votes
1answer
748 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
3
votes
2answers
376 views

Upper bound of |Aut(G)| for a p-group

If G is a p-group which is finitely generated with order p^n then what is the upper bound of |Aut(G)|.
2
votes
0answers
209 views

category theoretic approach to Sylow theorems and finite group theory?

Is there a category theoretic approach to Sylow theorems? More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...
2
votes
0answers
37 views
3
votes
1answer
572 views

Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
7
votes
1answer
108 views

Covering a set with images of a transversal

Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for ...
3
votes
3answers
278 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
13
votes
5answers
518 views

Results about the order of a group forcing a particular property.

Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property among all groups of ...
15
votes
5answers
2k views

Has any attempt been made to classify finite groupoids?

I recently stumbled upon the Mathieu groupoid and I found them fascinating. It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
-2
votes
0answers
132 views

About “covering” subgroups

Let $H$ be a subgroup of $S_n$ (the symmetric group with n elements). In the paper I read (cf. Thm 3.17 there), the authors define $H$ to be a covering if the following condition holds: for all ...
0
votes
1answer
94 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
3
votes
0answers
354 views

A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
4
votes
1answer
243 views

Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...
34
votes
20answers
6k views

Fun applications of representations of finite groups

Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
0
votes
1answer
89 views

Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer] My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...
6
votes
1answer
211 views

Finite field analogue of representations in same packet have equal central character

In Kevin Buzzard's recent question, a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal? Working my way ...
13
votes
1answer
462 views

Enumeration of $0-1$ matrices with determinant $1$

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$ been enumerated? E.g., for $n{=}2$, there are $f(2)=3$ such matrices: $$ \left( \begin{array}{cc} 1 & 0 \\ 0 ...
3
votes
0answers
160 views

Alternating quotients of (2,3,7;10)

It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...
3
votes
1answer
112 views

Finite p-groups and their fibered products

Is every finite $p$-group an epimorphic image of a fibered product of two finite $p$-groups which can be generated by $2$ elements?
4
votes
2answers
244 views

Maximal abelian subgroup of general linear groups

Thanks for any help or comments. Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$. By maximal abelian I mean if $A$ is ...
2
votes
1answer
139 views

Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...
3
votes
1answer
129 views

Subgroups of index 2 in a fibered product

Let $G$ be a finite group and let $M,N \lhd G$ be normal subgroups with a trivial intersection. Suppose that $G$ has a subgroup of index $2$. Must $G$ have a subgroup of index $2$ which contains ...
27
votes
2answers
1k views

How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
4
votes
1answer
166 views

About the set of Sylow-$p$ subgroups of $G$

Let G be a finite group and S be the set of Sylow p-subgroups of G for a prime p dividing the order of G. Assume that |S|>1. Let U and V be two disjoint non-empty subsets of S such that, ...
0
votes
0answers
59 views

Experimenting with the spider relator

The Monster group (actually the bimonster) has a presentation as Y555. Y555 is the quotient of a coxeter group (the coxeter diagram is a central node with three "spokes" coming out of it with length ...
1
vote
0answers
64 views

finite p-group subgroup of infinite p-group

is there any finite p-group G that is subgroup or minimal/maximal subgroup of infinite p-group H? if yes what is the limits? can this happen with different p's? i'm more interested in being maximal ...
1
vote
1answer
170 views

permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds. In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...
1
vote
1answer
223 views

Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors. A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...
3
votes
0answers
134 views

Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure. Fix ...
12
votes
2answers
923 views

(co)homology of symmetric groups

Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
0
votes
2answers
491 views

Similarity about unitary matrices

Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in ...
15
votes
2answers
745 views

In what sense is the classification of all finite groups “impossible”?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
2
votes
1answer
214 views

Irreducible constituent in a normal subgroup

Let $G$ be a finite group and $N$ be a normal subgroup of G. Suppose that $\chi \in Irr(G)$. If $\theta, \lambda \in Irr(N)$ satisfy $[\chi_{N}, \theta] > 0$ and $[\chi_{N}, \lambda] > 0$, is ...
10
votes
4answers
2k views

Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ? (Main case - complex numbers, comments on other cases are also welcome. "Given" ...
10
votes
0answers
211 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 ...
7
votes
1answer
375 views

Which finite groups are not the automorphism group of some rooted finite tree?

The question is as given in the title: Which finite groups are not the automorphism group of some rooted finite tree? A rephrasing could be: Is any finite group representable as the automorphism ...
0
votes
1answer
100 views

A bijection between Lusztig series induced by inflation

Context: Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F ...
3
votes
1answer
325 views

Reference for Clifford theory of algebras

Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II", Theorem 11.1. ...
1
vote
0answers
123 views

Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$ [closed]

Let $q=p^n$ for $p>3$.I want to know whether the group $G_2(q)$ has a subgroup of order $q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ...