Join operator

This is a binary subgroup property operator, viz an operator that takes as input two subgroup properties, and outputs one subgroup property

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Definition

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 $H=<K_{1},K_{2}>$.