Factors of Group Direct Product are not Subgroups

Theorem

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.

Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.

Then neither $\struct {G, \circ_1}$ nor $\struct {H, \circ_2}$ is a subgroup of $\struct {G \times H, \circ}$.

Proof

A subgroup is by definition a subset which is a group.

But neither $G$ nor $H$ are actually subsets of their cartesian product $G \times H$.

Hence the result.

$\blacksquare$