We develop the local-global theory of blocks for profinite groups. Given a field k characteristic p and group G, one may express completed algebra k[[G]] as product ∏i∈IBi closed indecomposable algebras, called G. To each block B G we associate pro-p subgroup defect B, unique up to conjugacy in give several characterizations analogy with groups finite Our main theorem is version Brauer's first ...

We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of the Galois representation ρE,p is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.

This note shows how to construct a representation of the automorphic extension G of the simple group S = O 8 (5) by a symmetric group on three points, together with an embedding of the normalizer H of an O 8 (2) type subgroup of O + 8 (5). As an application, it is shown that the permutation representation of G on the cosets of H has a base of length two. This question arose in [BGS11].

Let G be an arbitrary finite group and fix a prime number p. The McKay conjecture asserts that G and the normalizer in G of a Sylow p-subgroup have equal numbers of irreducible characters with degrees not divisible by p. The Alperin-McKay conjecture is version of this as applied to individual Brauer p-blocks of G. We offer evidence that perhaps much stronger forms of both of these conjectures a...

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert– Mumford–Kempf from geometric invariant theory. We ded...

Let Out(Fn) denote the outer automorphism group of the free group Fn with n > 3. We prove that for any finite index subgroup Γ < Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian : every injective homomorphism Γ → Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).

On the basis of an initial interest in symmetric cryptography, present work we study a chain subgroups. Starting from Sylow $2$-subgroup AGL(2,n), each term is defined as normalizer previous one group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values $n$, index consecutive does not depend $n$. Indeed, there strong evidence that sequence...

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with normalizer $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves $n$-torsion subgroup ideal class This is defined...

Let Γ be a nonelementary Kleinian group and H<Γ finitely generated, proper subgroup. We prove that if has finite covolume, then the profinite completions of H are not isomorphic. If index in Γ, there is onto which maps but does not. These results streamline existing proofs exist full-sized groups profinitely rigid absolute sense. They build on circle ideas can used to distinguish among subgroup...