Internal Direct Product Theorem/Proof 2

Theorem

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

Let $H_1, H_2 \le G$.

Then $G$ is the internal group direct product of $H_1$ and $H_2$ if and only if:

$(1): \quad G = H_1 \circ H_2$

$(2): \quad H_1 \cap H_2 = \set e$

$(3): \quad H_1, H_2 \lhd G$

where $H_1 \lhd G$ denotes that $H_1$ is a normal subgroup of $G$.

Proof

A specific instance of the general result, with $n = 2$.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Proposition $13.5$: Remark