## Non averaging sequences in finite groups

Let us say that a non averaging sequence in a group G is a sequence x1,…,xn such that x2i≠xjxk for any indices i,j,k such that two at least are distinct. My question is: given n, what is the smallest group having a non-averaging sequence of length n? Has this question been already tackled in the … Read more

## Characterization for special linear group over finite fields

Thanks for any help or comments. In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the following on page 56 of the book The Classification of the Finite Simple Groups by Gorenstein, Lyons and Solomon. Let G be a finite … Read more

## Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let G be a finite group,let Rep(G) denote the category of finite dimensional representations over C. Let V,W be representations of G in Rep(G). One can define a bilinear form on Rep(G) or inner product in K0(Rep(G)) (in Teleman’s notes) as dimCHom(V,W)G which is G invariant of Hom(V,W).Then there … Read more

## Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices. The (conjugate) Steinberg idempotent is defined to be $$e_n’=\frac{1}{q_n}\Sigma _{g\in \Sigma _n}sgn(g)g \Sigma _{g\in B _n}g \in Z_{(2)}[Gl_n(Z/2)]$$ where $q_n$ is an appropriate odd number, $sgn$ is the signature of the permutation. Now, let’s … Read more

## Lifting automorphisms of quotient groups

I am concerning here a natural question: Problem: Let G be a finite group, and let N be a characteristic subgroup of G. When can an automorphism φ∈Aut(G/N) be lifted to an automorphism of G? The problem in general seems difficult, and it has raised four years before in MathOverflow (see this). Question: for which … Read more

## Group of Lie type as expanders: explicit estimates

In a paper Finite simple groups as expanders by M. Kassabov, A. Lubotzky and N. Nikolov there is a theorem, stating that there exists ε>0 and k∈N, such that for every non-abelian finite simple group of Lie type (except the Suzuki group) its Cayley graph with respect to some k elements generating set is an … Read more

## When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?, $G$ must have a subgroup of the form $Z_p\times Z_p$ for some prime $p$. My question is: under what … Read more

## Given finite G⊂O(n)G\subset O(n), is there a “standard” cell structure on Sn−1S^{n-1} with GG acting cellularly?

Let G⊂O(n) be a finite orthogonal group. Is there a regular CW-complex structure on Sn−1 on which G acts cellularly which is in any sense “natural”? What I’m looking for is inspired by the following examples and ideally would generalize them: (a) If G is a reflection group, then the Coxeter complex, i.e. the regular … Read more

## the Yoneda embedding for finite groups and surjections from a free group

Let $F_2$ be the free group of rank 2, and let $G$ be a finite 2-generated group. To any such $G$, we may associate the set of surjections $$G(F_2) := Surj(F_2,G)$$ which admits a natural action of $Aut(F_2)$. Furthermore, any surjection $p : G\twoheadrightarrow G’$ induces a natural map $$p_* : \underbrace{Surj(F_2,G)}_{G(F_2)}\longrightarrow \underbrace{Surj(F_2,G’)}_{G'(F_2)}$$ which by … Read more

## Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity

For which triples {A,B,C} of positive integers does there exist a (finite or compact) group G with unitary irreducible representations of dimensions A,B, and C whose tensor product contains the trivial representation? Is this known in general? If not, what useful necessary and/or sufficient conditions are known? For example, it is necessary that A≤BC , … Read more