If $H \trianglelefteq G$ is a normal subgroup, a **complement** of $H$ is a subgroup $K \subseteq G$ with $\lvert H \cap K \rvert = 1$ and $HK=G$ where
$$HK = \{ hk \mid h\in H \text{ and } k\in K\}.$$

In order for a complement to exist, the sequence $$1 \to H \to G \to G/H \to 1$$ must split, in which case the complements are precisely the images of the possible splittings $G/H \to G$. All complements are isomorphic to $G/H$, and if $K$ is any complement then $G$ can be described as the internal semidirect product $H \rtimes K$.

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**Knowl status:**

- Review status: reviewed
- Last edited by Jennifer Paulhus on 2022-07-18 18:28:15

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**History:**(expand/hide all)

- 2022-07-18 18:28:15 by Jennifer Paulhus (Reviewed)
- 2022-07-02 17:12:48 by John Jones
- 2021-06-21 06:09:56 by David Roe

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