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 zeta (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/Omega(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 Omega(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 a Syl (p) (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]).