On the structure of groups whose exterior or tensor square is a p-group

It is well known that if G is a nilpotent (infinite) p-group of bounded exponent, then G⊗G (resp. G∧G) is also an (infinite) p-group. We study the converse under some restrictions. We also prove that if G is a finitely generated group with Gab capable, then the finiteness of G∧G implies that of G.