论文标题
主要电力订单组的子组一致性
Subgroup congruences for groups of prime power order
论文作者
论文摘要
给定一个$ p $ -group $ g $和子组关闭的类$ \ mathfrak {x} $，我们将每个$ \ m athfrak {x} $ - 子组$ h $ h $某些数量计算$ \ mathfrak {x} $ - 子集团 - 包含$ h $的子集团，这些$ h $受属于进一步的属性。我们在定理中显示，上述数量始终为$ \ equiv 1 \ pmod p $，并且仅当其他数量适合其他数量时。在定理II中，我们通过关注普通的$ \ mathfrak {x} $ - 子组来补充上述结果，在定理III中，我们获得了相对于阿伯利亚有限指数类别的著名定理的鲜明版本。还提出了其他各种推论。
Given a $p$-group $G$ and a subgroup-closed class $\mathfrak{X}$, we associate with each $\mathfrak{X}$-subgroup $H$ certain quantities which count $\mathfrak{X}$-subgroups containing $H$ subject to further properties. We show in Theorem I that each one of the said quantities is always $\equiv 1 \pmod p$ if and only if the same holds for the others. In Theorem II we supplement the above result by focusing on normal $\mathfrak{X}$-subgroups and in Theorem III we obtain a sharpened version of a celebrated theorem of Burnside relative to the class of abelian groups of bounded exponent. Various other corollaries are also presented.
