Factors of Group Direct Product are not Subgroups
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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.
$\blacksquare$