Intersections of $\Gamma$-isotype subgroups in abelian groups

Abstract:

A subgroup $H$ of an abelian group $G$ is an intersection of isotype subgroups of $G$ if and only if, for each prime $p$, if $x + H$ is a coset of order $p$ then there is another coset of order $p$ containing an element $y$ of order $p$ such that $h_p^* (x) \leqslant h_p^*(y)$. A subgroup $H$ of $G$ is isotype in $G$ if and only if, for each prime $p$, every coset $x + H$ of order $p$ contains an element y of order $p$ such that $h_p^*(x)h_p^*(y)$