Generalizations of Certain Elementary Theorems on p‐Groups
Generalizations of Certain Elementary Theorems on p‐Groups
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DOI：
10.1112/plms/s3-11.1.1
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发表时间：
1961
影响因子：
1.8
通讯作者：
N. Blackburn
中科院分区：
文献类型：
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作者：
N. Blackburn
本说明的目的是给这些定理一个更一般的背景。事实上，P.Roquette((9)，引理3)最近给出了(I)的推广，我们自己的过程也是沿着类似的路线进行的。我们将使用以下符号。对于任意有限^-群G，我们用Pk(G)表示由G的所有元素的Pk次方生成的子群，由O(Cr)表示G的Frattini子群。设d(G)是G的任何极小生成元集中的元素的个数，在G是正则的情况下，co(G)是G的不变量的个数。Ek(G)表示G的mostpk阶元的集合；在正则情况下Ek{G)是一个子群，ex{G)是P0*^阶的。如果a，y是G的元素，我们用公式定义交换子[x，y]和变换XV：[x，y]=x-xy-xxy，x^=x[x，y]=y~xxy。
The aim of the present note is to give a more general context to these theorems. A generalization of (I) was in fact recently given by P. Roquette ((9), Lemma 3), and our own procedure is along similar lines. We shall use the following notation. For any finite^>-group G we denote by Pk (G) the subgroup generated by the pk-th powers of all elements of G and by O (Cr) the Frattini subgroup of G. Let thus d (G) is the number of elements in any minimal set of generators of G, and in the case when G is regular co (G) is the number of invariants of G. Ek (G) denotes the set of elements of G of order at mostpk; in the regular case Ek {G) is a subgroup and EX {G) is of order p0*^. If a;, y are elements of G we define the commutator [x, y] and the transform xv by the formulae:[x, y]= x-xy-xxy, x^= x [x, y]= y~ xxy.