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