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In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvable . Further progress was made showing that CN-groups, groups in which the centralizer of a non-identity element is nilpotent, of odd order are solvable . The complete solution was given in , but further work on CN-groups was done in , giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group ''G'' is such that its largest solvable normal subgroup ''O''∞(''G'') is a 2-group, and the quotient is a group of even order. ==Examples== Solvable CN groups include *Nilpotent groups *Frobenius groups whose Frobenius complement is nilpotent *3-step groups, such as the symmetric group ''S''4 Non-solvable CN groups include: *The Suzuki simple groups *The groups PSL2(F2''n'') for ''n''>1 *The group PSL2(F''p'') for ''p''>3 a Fermat prime or Mersenne prime. *The group PSL2(F9) *The group PSL3(F4) 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「CN-group」の詳細全文を読む スポンサード リンク
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