21.137 (2026)

If the $p$-th powers in a finite $p$-group form a subgroup, must that subgroup be powerful? That is, for $p \neq 2$, if the $p$-th powers in a $p$-group of exponent $p^2$ form a subgroup, must that subgroup be abelian? For a 2-group of exponent 8, if the squares form a subgroup, must that subgroup be abelian?