Isolated elements of prime order in finite groups
Isolated elements of prime order in finite groups
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有限群中素数阶的孤立元素
DOI：
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发表时间：
1988
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通讯作者：
O. D. Artemovich
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作者：
O. D. Artemovich
设G是有限群，p是划分其阶的素数，P是G的Sylow p子群，x是P的p阶元，如果x((X))不与P{x}中的任何元素(P{r}中的任何子群)共轭，则称元素x(子群)关于G在P中是孤立的.Glauberman[I]证明了当p=2时，群G的孤立对合且每个零次正规子群为偶数阶的孤立对合必属于中心Z(G)。他提出了以下问题：奇素数p的Z*-定理的类比是否有效(见[I，2；问题4.21])7
Let G be a finite group, p be a prime number that divides its order, P be a Sylow psubgroup of G, and x be an element of order p of P. The element x (the subgroup ) is said to be isolated in P with respect to G if x ((x)) is not conjugate in G to any element of P {x} (to any subgroup in P {r} ). Glauberman [i] has proved the Z*-theorem that for p = 2 an isolated involution of the group G, each n0ntrivial normal subgroup of which has even order, always belongs to the center Z(G). He posed the following problem: Is the analogue of the Z*-theorem for odd prime numbers p valid (see [I, 2; Problem 4.21])7