More than 70 years ago, P. Hall showed that if G is a finite p -group such that a term G(d+1) of the derived series is non-trivial, then the order of the quotient G(d)/G(d+1) is at least p2 +1 . Recently Mann proved that, in a finite p -group, Hall’s lower bound can be taken for at most two distinct d . For odd p , we prove a sharp version of this result and characterise the groups with two sma...

We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite p-group G. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup H of G. When G is a semidirect product of two finite p-groups H and K, both H and K admit Gray maps and the corresponding homomorphism ψ : H −→ Aut(K) is compatible with the Gray map of K in a sen...

It would be interesting to extend this result by allowing B to have nilpotence class 2 instead of necessarily being abelian. This cannot be done if p = 2 (Example 4.2), but perhaps it is possible for p odd. (It was done by the author ([Gor, p.274]; [HB, III, p.21]) for the special case in which p is odd and [B,B] ≤ A.) However, there is an application of Thompson’s Replacement Theorem that can ...

This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presen...