Finite p-central groups of height k

A finite group G is called p^i-central of height k if every element of order p^i of G is contained in the k^th-term ζ_k(G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω_1(P) is also p-central of height p − 2 (Thm. B). In such a group P , the index of P^p is less than or equal to the order of the subgroup Ω_1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p-1, p odd, and N_G(P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Sylp(G) is p- central of height p−2, then N_G (P ) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11])