On p-groups whose L-automorphism group is transitive on the atoms (Q1117031)

An L-isomorphism of a group G onto a group H is an isomorphism of the subgroup lattice of G onto that of H. It was proved by Shult that if the automorphism group of a p-group G (p odd) is transitive on the set of subgroups of order p, then G is Abelian, and on the basis of this, the author conjectures that if the L-automorphism group of a p-group G is transitive on the set of subgroups of order p, then G is modular. Continuing a previous study of this question, it is proved that G is modular if the L-automorphism group has a subgroup H of order prime to p which acts transitively on the set of subgroups of order p. It is a consequence that the same holds if H is assumed to be soluble rather than of order prime to p.