SYLOW p-SUBGROUPS OF THE CLASSICAL GROUPS OVER FINITE FIELDS WITH CHARACTERISTIC PRIME TO p
SYLOW p-SUBGROUPS OF THE CLASSICAL GROUPS OVER FINITE FIELDS WITH CHARACTERISTIC PRIME TO p
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DOI：
10.1090/s0002-9939-1955-0072143-9
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发表时间：
1955-04
期刊：
影响因子：
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通讯作者：
A. J. Weir
中科院分区：
文献类型：
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作者：
A. J. Weir
如果S是pn次对称群的Sylow p-子群，则任何pn阶群都可以嵌入到Sn中。我们可以将锡表示为完整的产物‘CO，C，O…本文的目的是证明有限域GF(Q)上具有q个元素的经典群(见？1)的任一Sylow p子群，其中(q，p)=1，可表示为基本子群En-COCoO的直积。O C(n因子)，其中Z是Pr阶循环。(我们总是假设p；2。)由于C可以嵌入到S中，所以我们看到n以一种特别简单的方式嵌入到了Snr-L中。上述r由方程式q-1=pt*定义，其中qi是q的第一次幂，它与1mod p同余，*表示某个未指定的p的素数。因此，r=1的情况经常发生，而且很明显，SnS教授菲利普·霍尔是我1949-1952年间在剑桥(英国)的研究导师，我很高兴在此感谢他的慷慨鼓励。
If S,, is a Sylow p-subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in Sn. We may express Sn as the complete product' C o C o ... o C of n cyclic groups of order p and the purpose of this paper is to show that any Sylow psubgroup of a classical group (see ?1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups En-C O C O ... o C (n factors), where Z is cyclic of order pr. (We assume always that p ;2.) Since C may be imbedded in S., we see that n is imbedded in Sn+r-l in a particularly simple way. The above r is defined by the equation q -1 =pt *where qI is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly SnSn Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowledge here my indebtedness to him for his generous encouragement.