Let G be a group generated by a finite set A. An element g ∈ G is a strict dead end of depth k (with respect to A) if |g| > |ga1| > |ga1a2| > · · · > |ga1a2 · · · ak| for any a1, a2, . . . , ak ∈ A such that the word a1a2 · · ·ak is freely irreducible. (Here |g| is the distance from g to the identity in the Cayley graph of G.) We show that in finitely generated free soluble groups of degree d ≥...

Abstract The aim of this paper is to provide a contribution the theory uncountable groups and that paranilpotent groups. Extending structural results in Franciosi de Giovanni (Ricerche Mat 40:321–333, 1991) et al. (Comm Algebra 49:3020–3033, 2021), we prove locally soluble minimal non-paranilpotent , i.e. whose proper subgroups are paranilpotent, soluble. It also shown class countably recogniza...

We show that for soluble groups of type FPn, centralisers of finite subgroups need not be of type FPn.