Further research problems and theorems on prime power groups

Abstract

4101. Describe the p-groups all of whose subgroups of index p, k ∈ {2, 3, 4}, are normal (three problems). Consider in detail the groups of exponent p. 4102. Study the nonabelian p-groups G all of whose maximal abelian subgroups are normal (any two elements of G generate a subgroup of class ≤ 2 so our group is regular if p > 2, by Theorem 7.1(b) in [B1]). Consider in detail the case p = 2. 4103. Find the maximal possible order of the automorphism groups of the groups of maximal class of order p. 4104. Study the non-Dedekindian p-groups covered by nonnormal subgroups. 4105. Study the p-groups G in which the intersection of any two nonincident subgroups, say A and B, of equal order (of different orders) is normal (i) either in A or in B, (ii) in 〈A,B〉. 4106. Study the p-groups G all of whose nonabelian subgroups of equal order are isomorphic (permutable). Consider in detail the case when exp(G) = p.