LAUSR

A finite group G is called $$\psi $$ ψ -divisible if $$\psi (H)|\psi (G)$$ ψ ( H ) | ψ ( G ) for any subgroup H of G , where $$\psi (H)$$ ψ ( H ) and $$\psi (G)$$ ψ ( G ) are the sum of element orders of H and G , respectively. In this paper, we classify the finite groups whose subgroups are all $$\psi $$ ψ -divisible. Since the existence of $$\psi $$ ψ -divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $$\psi $$ ψ -divisibility property of ZM-groups. In the end, we introduce the concept of $$\psi $$ ψ -normal divisible group, i.e., a group for which the $$\psi $$ ψ -divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $$\psi $$ ψ -normal divisible groups which are neither simple nor nilpotent.