Search

The weak commutativity group χ ⁢ (G) is generated by two isomorphic groups 퐺 and G φ subject to the relations [ g , g φ ] = 1 for all g ∈ G . We present new bounds for the exponent of χ ⁢ (G) and its sections, when 퐺 is a finite 푝-group. [ABSTRACT FROM AUTHOR]

We prove that the kth term of the lower central series of a finite group G is nilpotent if and only if |ab| = |a| |b| for any γk-commutators a, b ∈ G of coprime orders. [ABSTRACT FROM AUTHOR]

Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). In this paper the finite non-solvable groups G with ω(G)≤6 are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite non-solvable groups G with ω(G)=7. Moreover, it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω(G)≤n, generalizing a result of Kohl. [ABSTRACT FROM AUTHOR]