If $G$ is a group with normal subgroup $N$ and quotient $Q\cong G/N$, then $G$ is a **extension** of $Q$ by $N$, denoted $N.Q$. Equivalently there is a short exact sequence
$$1 \to N \to G \to Q \to 1.$$

If $G$ has a subgroup $\tilde Q\leq G$ such that $\tilde Q\cap N=\langle e\rangle$, and $\tilde QN=G$, then $\tilde Q\cong Q$ and $G$ is a semidirect product of $N$ and $\tilde Q$. This is equivalent to the existence of a section of the quotient map $G \to Q$. In this case we say $G$ is a **split extension** of $Q$ by $N$.

If no such subgroup $\tilde Q$ exists, then $G$ is a **non-split extension** of $Q$ by $N$.

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**Knowl status:**

- Review status: beta
- Last edited by Jennifer Paulhus on 2022-07-18 18:25:47

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

- 2022-07-18 18:25:47 by Jennifer Paulhus
- 2022-07-18 18:12:14 by Jennifer Paulhus
- 2020-12-08 01:08:58 by David Roe
- 2019-06-12 14:47:17 by John Jones

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