In 2000, L. Héthelyi and B. Külshammer proved that if p is a prime number dividing the order of a finite solvable group G, then G has at least 2 √ p − 1 conjugacy classes. In this paper we show that if p is large, the result remains true for arbitrary finite groups.

We provide an example of a non-finitely generated group which admits nonempty strongly aperiodic subshift finite type. Furthermore, we completely characterize the groups with this property in terms their finitely subgroups and roots conjugacy classes.

Let G be a pro-p group of finite cohomological dimension and type FP∞ and T is a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP∞ (in particular it is finitely presented). Moreover we prove that a pro-p group G of type FP∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.