Join operator

Definition with symbols
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 $$H = $$.