Subgroup Subset of Subgroup Product

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $H$ and $K$ be subgroups of $G$.

Then:

$H \subseteq H \circ K \supseteq K$

where $H \circ K$ denotes the subset product of $H$ and $K$.

Proof

By definition of subset product:

$H \circ K = \set {h \circ k: h \in H, k \in K}$

So:

\(\ds x\)

\(\in\)

\(\ds H\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds x \circ e\)

Definition of Identity Element

\(\ds \)

\(\in\)

\(\ds H \circ K\)

Identity of Subgroup

\(\ds \leadsto \ \ \)

\(\ds H\)

\(\subseteq\)

\(\ds H \circ K\)

Definition of Subset

and:

\(\ds y\)

\(\in\)

\(\ds K\)

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds e \circ y\)

Definition of Identity Element

\(\ds \)

\(\in\)

\(\ds H \circ K\)

Identity of Subgroup

\(\ds \leadsto \ \ \)

\(\ds K\)

\(\subseteq\)

\(\ds H \circ K\)

Definition of Subset

$\blacksquare$