Finite p-groups whose nonnormal subgroups are metacyclic

For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.

On the construction of irreducible characters in p-blocks with normal metacyclic defect groups

The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducible ordinary and modular characters in this type of blocks.As a by-product,k(B) and l(B) are obtained.

Finite 2-generator equilibrated p-groups

Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note.

A Characterization of the Smallest Suzuki 2-Group

Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.