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In mathematics, George Glauberman's Z * theorem is stated as follows: Z This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer-Suzuki theorem to deal with some small cases). The original paper 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 A simple corollary is that an element ''t'' in ''T'' is not in ''Z *''(''G'') if and only if there is some ''s'' ≠ ''t'' such that ''s'' and ''t'' commute and ''s'' and ''t'' are ''G'' conjugate. A generalization to odd primes was recorded in : if ''t'' is an element of prime order ''p'' and the commutator (''g'' ) has order coprime to ''p'' for all ''g'', then ''t'' is central modulo the ''p''′-core. This was also generalized to odd primes and to compact Lie groups in , which also contains several useful results in the finite case. have also studied an extension of the Z * theorem to pairs of groups ''(G,H)'' with ''H'' a normal subgroup of ''G''. ==References== * gives a detailed proof of the Brauer–Suzuki theorem. * * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Z* theorem」の詳細全文を読む スポンサード リンク
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