M.Sc Thesis


M.Sc StudentBerman Stav
SubjectPermutations of Cosets
DepartmentDepartment of Mathematics
Supervisor DR. Nir Lazarovich


Abstract

We study symmetries of a group, which are manifested as permutations of cosets of finite-index subgroups. We realize them in two ways. The first is the so-called profinite full group of a finitely generated residually finite group. In existing terms, it can be defined as the topological full group of a group's action on its profinite completion. We describe it in elementary terms as a certain group of permutation of a finitely generated residually finite group. Namely permutations of a group which restrict to a left translations on the cosets of some finite-index subgroup. Borrowing techniques from Bazouglyi-Medynets we show its derived subgroup is simple, and that it has only abelian quotients. We provide tools that can be used in some cases to describe its abelianization.


The other type of coset symmetry we describe are so-called coset automoprhisms. These are simply certain order automorphisms of the poset of finite-index subgroup cosets, of a finitely generated group G. We require that our coset automorphism preserve the algebraic structure of G in some sense, namely we require that they preserve its finite quotients. We distinguish between two variants coset automorphisms: those which are induced by a permutation of the group, and those which are not. Only the former acts on G, but it turns out that symmetries of the latter variety do have an action of the profinite completion Ĝ.


There has been work in the past about symmetries similar to the first variant, only as automorphisms of the poset of all subgroup cosets, including infinite index. For the first variant we generalize some existing results to our finite-index setting. In the second variant, we compute some specific examples - including for free groups and abelian groups - giving a description in terms of the profinite completion.