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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], and let $O(G)$ be the largest [[normal subgroup|0CB0DX5BVV]] of $G$ of odd order.\nLet $T$ be a [[Sylow $2$-subgroup|0053CQE9Y2]] of $G$, and let $t \\in T$ be an involution, that is, an element of [[order|0WE2NHZJ7B]] $2$.\nSuppose that no element of $T$ other than $t$ is [[conjugate|0DXD4QSZ8A]] to $t$ in $G$.\nThen $t \\, O(G)$ lies in the [[centre|0DZ93493VN]] of the [[quotient group|0KYX2N69J0]] $G/O(G)$.\n\n## Remarks\n\nThe theorem is due to George Glauberman, and its proof uses modular representation theory, in particular Brauer's theory of blocks.\nThe notation comes from $Z^*(G)$, the subgroup of $G$ containing $O(G)$ with $Z^*(G)/O(G) = Z(G/O(G))$; the conclusion says $t \\in Z^*(G)$.\n\nExample: $G = S_3$ and $t = (1\\,2)$ in $T = \\set{e, (1\\,2)}$.\nNo other element of $T$ is conjugate to $t$, and $O(S_3) = A_3$.\nIndeed $S_3 / A_3$ has order $2$, so it is abelian and $t A_3$ is central.\n\nCorollary: if $G$ is a nonabelian [[simple group|0YMYMNQ0QZ]], then every involution $t$ in a Sylow $2$-subgroup $T$ is conjugate in $G$ to another element of $T$.\nIndeed, $G$ has even order because it contains $t$, so $O(G) \\ne G$; hence $O(G) = \\set{e}$ by simplicity, and the centre of $G$ is trivial, so $t$ cannot be central.\n\n## Explanations\n\n### L1 human\n\nAn involution is an element that undoes itself, like a reflection.\nCall an involution isolated if conjugation never moves it onto another element of its Sylow $2$-subgroup.\nThe Z* theorem says that an isolated involution becomes central after dividing out the largest normal subgroup of odd order.\nSo isolated involutions cannot exist in a nonabelian simple group.\n\n### L2 human\n\nThe Z* theorem is a result about fusion of $2$-elements, like the [[focal subgroup theorem|H1BSG0J5G1]] but for single involutions.\nIt was an important tool in the classification of finite simple groups, where it rules out isolated involutions.\nThe proof by modular character theory is a striking case where representation theory is needed for a statement about conjugacy in a group.\n",
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