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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 $N.Q$. 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 $N$.

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