论文标题
Butler的方法适用于$ \ Mathbb {Z} _p [C_P \ times C_P] $ - 排列模块
Butler's Method applied to $\mathbb{Z}_p[C_p\times C_p]$-permutation modules
论文作者
论文摘要
令$ g $为有限的$ p $ group，带有普通亚组$ n $ $ p $。第一作者和Zalesskii以前曾在$ g/n $的模块中对$ \ mathbb {z} _pg $的排列模块进行了描述，但其条件的必要性尚不清楚。我们对巴特勒（Butler）的信函应用来证明条件的必要性，通过向高度非平凡的反例展示，如果$ n $ invariants和$ n $ invariants和给定晶格$ u $的$ n $ coinvariants都是置换模块，则是$ u $。
Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first author and Zalesskii have previously given a characterization of permutation modules for $\mathbb{Z}_pG$ in terms of modules for $G/N$, but the necessity of their conditions was not known. We apply a correspondence due to Butler to demonstrate the necessity of the conditions, by exhibiting highly non-trivial counterexamples to the claim that if both the $N$-invariants and the $N$-coinvariants of a given lattice $U$ are permutation modules, then so is $U$.
