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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 $N.Q$.

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 $G \to Q$.

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

Knowl status:
  • Review status: beta
  • Last edited by David Roe on 2020-12-08 01:08:58
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