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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]].\nThe *Fitting subgroup* $F(G)$ is the subgroup [[generated|0RP7VWQ0YN]] by all [[normal subgroups|0CB0DX5BVV]] of $G$ that are [[nilpotent|0Q7MMQHC1P]].\n\n## Remarks\n\nFitting's theorem says that if $M$ and $N$ are nilpotent normal subgroups of $G$, then $MN$ is a nilpotent normal subgroup.\nHence $F(G)$ is itself nilpotent and normal, and it contains every nilpotent normal subgroup of $G$: it is the unique largest one.\n$F(G)$ is a [[characteristic subgroup|0KABPMRHM2]], and it is the product over the primes $p$ of the largest normal $p$-subgroups $O_p(G)$.\nIf $G$ is [[solvable|0FXJ7YFMMX]], then the [[centraliser|02YYV36BFM]] of $F(G)$ in $G$ is contained in $F(G)$.\n\nExample: $G = S_4$.\nIts normal subgroups are $\\set{e}$, $V = \\set{e, (1\\,2)(3\\,4), (1\\,3)(2\\,4), (1\\,4)(2\\,3)}$, $A_4$ and $S_4$.\nThe group $V$ is abelian, hence nilpotent, and it is the [[Klein four-group|0PMRQJVTME]].\nThe groups $A_4$ and $S_4$ are not nilpotent, because their [[centres|0DZ93493VN]] are trivial.\nSo $F(S_4) = V$.\n\nNon-example: $A_4$ is a normal subgroup of $S_4$ that is not nilpotent, so it is not contained in the generating family, and it is not $F(S_4)$.\n\nIf $G$ is a nonabelian [[simple group|0YMYMNQ0QZ]], then $F(G) = \\set{e}$.\nIf $G$ is nilpotent, then $F(G) = G$.\n",
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  "name": "Fitting subgroup",
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