Finite groups all of whose small subgroups are normal

Abstract

The finite groups G such that, whenever H < G and |H|2 < |G|, then H ◃ G are classified in Theorems 2.1 and 3.2. Let pν(X)(pa(X)) be the maximal order of an abelian (a minimal nonabelian) subgroup of a p-group X. We classify the p-groups G such that, whenever an abelian H < G and |H| ≤ pν(G)-2, then H ◃ G. Next, we describe the p-groups G all of whose abelian subgroups of order ≤ pa(G)-2 are normal. Minimal nonabelian and minimal nonnilpotent groups play crucial role in the whole paper.