High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, George Glauberman's Z* theorem states that if G is a finite group and T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G). The subgroup Z*(G) is the inverse image in G of the center of G/O(G), where O(G) is the maximal normal subgroup of G of odd order. This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer-Suzuki theorem to deal with some small cases). The original paper (Glauberman 1966) gave several criteria for an element to lie outside Z*(G). Its theorem 4 states: For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties: 1. g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) NG(CT(U)). 2. t is contained in U and tg gt. 3. U is generated by the N-conjugates of t. 4. the exponent of U is equal to the order of t.

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