Join operator

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

Template:Property existentialization

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Definition

Definition with symbols

Given two subgroup properties $p$ and $q$ , the join of $p$ and $q$ , denoted as $⟨ p, q ⟩$ , is defined as follows: a subgroup $H$ satisfies property $⟨ p, q ⟩$ 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 $H = < K_{1}, K_{2} >$ .