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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
82 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 ...
0
votes
1answer
123 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 ...
12
votes
2answers
840 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 ...
3
votes
1answer
529 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 ...
0
votes
2answers
467 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 ...
3
votes
0answers
197 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: By applying this result to finite groups, we get a Galois correspondence ...
7
votes
1answer
97 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 ...
15
votes
2answers
704 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
208 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" ...
9
votes
0answers
184 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
351 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
86 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
321 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
110 views

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

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 ...
2
votes
1answer
144 views

What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
1
vote
0answers
72 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
2answers
240 views

If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...
3
votes
1answer
101 views

Covering finite groups by kernels

Let $G$ be a finite group. When does there exist a finite group $H$ such that every $h\in H$ is in the kernel of some epimorphism $H\to G$? This is well-known to be true for $G$ abelian, for example ...
-4
votes
0answers
55 views

izomorphism of finite abelian group [closed]

Please help me with rezolving this problem from Romanian "Gazeta Matematica" : "an finite abelian group G have |End G | and |Aut G | coprime numbers. Show that |G| is square free. Thank you!
-1
votes
0answers
76 views

Represention of the element of a group: A Confusion [closed]

I am reading a paper "FAST ALGORITHMS FOR CALCULATION OF GIBBS DERIVATIVES ON FINITE GROUPS" by R. S. Stankovic (Approx. Theory & its AppL 7:2, June 1991). In Section 2, following is written about ...
7
votes
2answers
447 views

Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups. For example, Artin's theorem is the statement that for every character $\chi$ of ...
4
votes
0answers
92 views

Improvements of the Reidemeister-Schreier index formula for particular classes of groups

I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then $$d(H) \le (d-1) ...
14
votes
1answer
328 views

Number of solutions to equations in finite groups

Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$. Is it always true that the number ...
2
votes
1answer
98 views

Units in a finite semisimple group algebra

Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...
1
vote
0answers
83 views

Generalization of groups with non-connected prime graph

Suppose $G$ is a non-abelian finite group and $\pi(G)=\pi_1(G)\cup \pi_2(G)$ is a disjoint union of all primes dividing $|G|$. Suppose further that for every two elements $g_1,g_2\in G\setminus Z(G)$ ...
2
votes
0answers
246 views

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

This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...
-1
votes
0answers
49 views

Quotient of Euclidean space under the action of a permutation group

Suppose that $G$ is a permutation group acting on the Euclidean space $\mathbb{R}^n$. I am interested in the geometric properties of the quotient space $\mathbb{R}/G$. Since $G$ is finite and the ...
1
vote
1answer
288 views

cohomology ring of symmetric group of order $3$

Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring $$ H^*(S_3;\mathbb{Z})?$$ My attempt: I want to use mathematical induction on $n$ for $S_n$. For $n=1$, $S_1$ is trivial. ...
8
votes
2answers
857 views

Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
2
votes
1answer
189 views

Structure of symplectic group over finite fields

We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of ...
18
votes
3answers
944 views

Groups of order $n$ with a character whose degree is at least $0.8\sqrt{n}$ (say)

[edited in response to some corrections by Geoff Robinson and F. Ladisch] Throughout, all my groups are finite, and all my representations are over the complex numbers. If $G$ is a group and $\chi$ ...
8
votes
2answers
339 views

Can any finite lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Question: Can any finite lattice be realized as ...
6
votes
2answers
230 views

Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group

Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$. What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...
3
votes
2answers
301 views

symmetric 2-cocycle / many projective representations

Let $G$ be a finite group, $k$ the field of complex numbers. Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$: ...
2
votes
0answers
51 views

explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$

I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...
0
votes
1answer
132 views

Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$. The case in which $p=2$ is well known, see e.g. Table 1 at ...
4
votes
1answer
119 views

Can any finite distributive weighted lattice be realized by inclusion of groups?

By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice. A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
11
votes
3answers
991 views

Number of generators of a subgroup of a finite simple group

For a finite group $G$ we denote $d(G)$ the minimal size of a set of generators of $G$. We define $D(G) = \max( d(H) \mid H\leq G)$. Let $S$ be a finite simple group. Are there `good' bounds on ...
3
votes
4answers
990 views

General bound for the number of subgroups of a finite group

I am interested in the following: Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that $|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ ...
3
votes
1answer
165 views

Centralizers in the universal central extensions of the alternating groups?

For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
19
votes
1answer
532 views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
0
votes
1answer
100 views

The simple groups with an absolutely irreducible projective representations with small degrees

In the question "Simple Hurwitz Groups of order less than 10^7", it came up that all the Hurwitz groups with absolutely irreducible projective representations of degrees up to 7 over any field are ...
5
votes
1answer
343 views

Simple Hurwitz Groups of order less than 10^7

I'm trying to calculate a table of all simple hurwitz groups of order less than 10^7. None of the tables I found went further than 10^6, so I decided to use the tables of all simple groups up to 10^7 ...
0
votes
0answers
78 views

Asymmetry of functions defined on the $n$-th roots of the unity

Let $\mathcal{A} = \{V : \mathbb{U}_n \rightarrow \mathbb{C}\}$ where $\mathbb{U}_n$ is the group of the complex $n$-th roots of the unity. This group naturally acts on $\mathcal{A}$: for any $a \in ...
10
votes
4answers
1k views

Reference for this theorem in representation theory?

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). If $H$ is the kernel of $\chi$ then the irreducible representations of ...
2
votes
1answer
382 views

Inseparable Galois Cohomology

First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
5
votes
0answers
149 views

Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...
6
votes
1answer
273 views

Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...