Subset Product is Subset of Generator

Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $X, Y \subseteq \left({G, \circ}\right)$.

Then $X \circ Y \subseteq \left \langle {X, Y} \right \rangle$ where:

• $X \circ Y$ is the Subset Product of $X$ and $Y$ in $G$.
• $\left \langle {X, Y} \right \rangle$ is the group generated by $X$ and $Y$.

Proof

It is clear from Set of Words Generates Group that $W \left({\hat X \cup \hat Y}\right) = \left \langle {X, Y} \right \rangle$.

It is equally clear that $X \circ Y \subseteq W \left({\hat X \cup \hat Y}\right)$.

$\blacksquare$