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  "body": "\n## Definition\n\nLet $N$ and $Q$ be [[groups|G5S1M9]].\nAn *extension* of $Q$ by $N$ is a group $G$ together with two [[group homomorphisms|0JYTG5K9Y4]] $\\iota \\colon N \\to G$ and $\\pi \\colon G \\to Q$ such that:\n\n1. $\\iota$ is [[injective|09NWETVTNY]].\n2. $\\pi$ is surjective.\n3. The [[image|0S21GNW9ZB]] of $\\iota$ equals the kernel of $\\pi$.\n\nIn other words, $1 \\to N \\xrightarrow{\\iota} G \\xrightarrow{\\pi} Q \\to 1$ is a short exact sequence of groups.\nThe extension is *split* if there is a homomorphism $s \\colon Q \\to G$ with $\\pi \\circ s = \\id_Q$.\nThe extension is *central* if $\\iota(N)$ lies in the [[centre|0DZ93493VN]] of $G$.\n\n## Remarks\n\nIn an extension, $\\iota(N)$ is a normal subgroup of $G$ and $G / \\iota(N) \\cong Q$.\nSome authors call the same object an extension of $N$ by $Q$, so the order of the words varies between books.\n\nExample: $G = S_3$, $N = A_3$ with the inclusion $\\iota$, and $Q = \\set{\\pm 1}$ with $\\pi$ the sign map.\nThe inclusion is injective, the sign map is onto, and the kernel of the sign map is $A_3$.\nThe extension is split: $s(-1) = (1\\ 2)$ is a homomorphism with $\\pi \\circ s = \\id$.\n\nNon-example: $\\iota \\colon \\Z \\to \\Z$, $n \\mapsto 4n$, and $\\pi \\colon \\Z \\to \\Z/2\\Z$, reduction modulo $2$.\nClauses 1 and 2 hold, but the image $4\\Z$ of $\\iota$ is not the kernel $2\\Z$ of $\\pi$, so clause 3 fails.\n\nThe 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 $2\\Z/4\\Z$.\nA split extension is the same as a [[semidirect product|0H3WZXCZNF]] $N \\rtimes Q$.\nWhen $N$ 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 $H^2(Q, N)$.\n",
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