Internal Direct Product Theorem/Proof 2
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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