Abstract
We determine the non-abelian finite p-groups G with CG(x) ⩽ H for any minimal non-abelian subgroup H of G and each x ∈ H – Z(G) (Theorem 1.1). This solves Problem 757 of Berkovich [Groups of prime power order, vol. 1 and vol. 2, 2008].
We also classify the non-abelian finite p-groups G such that whenever A is a maximal subgroup of any minimal non-abelian subgroup H in G, then A is also a maximal abelian subgroup in G (Theorem 1.2), and this solves another problem of Berkovich [Groups of prime power order, vol. 1 and vol. 2, 2008].
Finally, we generalize a result of Blackburn [J. Algebra 3: 30–37, 1966] concerning finite p-groups in which the non-normal subgroups have non-trivial intersection (Theorem 1.3 and Corollary 1.4).
© de Gruyter 2009