Group extension

Remarks

In an extension, $\iota(N)$ is a normal subgroup of $G$ and $G / \iota(N) \cong Q$. Some authors call the same object an extension of $N$ by $Q$, so the order of the words varies between books.

Example: $G = S_3$, $N = A_3$ with the inclusion $\iota$, and $Q = \set{\pm 1}$ with $\pi$ the sign map. The inclusion is injective, the sign map is onto, and the kernel of the sign map is $A_3$. The extension is split: $s(-1) = (1\ 2)$ is a homomorphism with $\pi \circ s = \id$.

Non-example: $\iota \colon \Z \to \Z$, $n \mapsto 4n$, and $\pi \colon \Z \to \Z/2\Z$, reduction modulo $2$. Clauses 1 and 2 hold, but the image $4\Z$ of $\iota$ is not the kernel $2\Z$ of $\pi$, so clause 3 fails.

The extension $0 \to 2\Z/4\Z \to \Z/4\Z \to \Z/2\Z \to 0$ does not split, because the only element of order $2$ in $\Z/4\Z$ lies in $2\Z/4\Z$. A split extension is the same as a semidirect product $N \rtimes Q$. When $N$ is abelian, the extensions of $Q$ by $N$ that induce a given action of $Q$ on $N$ are classified up to equivalence by the second cohomology group $H^2(Q, N)$.