Finite groups with submultiplicative spectra

We study abstract finite groups with the property, called property , that all of their subrepresentations have submultiplicative spectra. Such groups are necessarily nilpotent and we focus on p-groups. p-groups with property  are regular. Hence, a 2-group has property  if and only if it is commutative. For an odd prime p, all p-abelian groups have property , in particular, all groups of exponent p have it. We show that a 3-group or a metabelian p-group (p≥5) has property  if and only if it is V-regular.