factorizing profinite groups into two abelian subgroups

Maximal abelian subgroups of free profinite groups

THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ^ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then foll...

Triple factorization of non-abelian groups by two maximal subgroups

The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...

Maximal Subgroups in Finite and Profinite Groups

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann....

Counting the Closed Subgroups of Profinite Groups

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

Random Normal Subgroups of Free Profinite Groups