Active sums II

Abstract

We exhibit several finite groups that are not active sums of cyclic subgroups. We show that this is the case for groups with $H_{1}G$ of odd order and $H_{2}G$ of even order. As particular examples of this we have the alternating groups $A_n$ for $n\geq 4$, some special and some projective linear groups. Our next set of examples consists of $p$-groups where the normalizer and the centralizer of every element coincide. We also have an example of a 2-group where the above conditions are not satisfied; thus we had to devise an ad hoc argument. We observe that the examples of $p$-groups given also provide groups that are not molecular.