M.Sc Thesis | |

M.Sc Student | Berman Stav |
---|---|

Subject | Permutations of Cosets |

Department | Department of Mathematics |

Supervisor | DR. Nir Lazarovich |

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.