# Join operator

Given two subgroup properties $p$ and $q$, the join of $p$ and $q$, denoted as $\langle p,q \rangle$, is defined as follows: a subgroup $H$ satisfies property $\langle p, q \rangle$ in $G$ if there exist subgroups $K_1,K_2$ of $H$ such that $K_1$ satisfies $p$ in $G$, $K_2$ satisfies $q$ in $G$, and $H = $.