AI 中文总结
介绍有限群的团复形\(X_G\)，其\(1\) - 骨架以\(\Gamma_G\)为商，有限单群由其确定到同构。团复形可赋予序层，通过顶点链同调计算任意单纯复形同调，还计算了某些特殊群的序层系数团复形同调。
AI 中文摘要
我们引入有限群的团复形\(X_G\)：一个单纯复形，其面与群元素一一对应，通过将群元素分解为素数\(p\)的\(p\) - 元素得到。该复形的\(1\) - 骨架以经典的格鲁恩贝格 - 凯格尔图\(\Gamma_G\)作为商，且有限单群由其团复形确定到同构。团复形还可赋予一个组合层——称为序层——并且可根据顶点链的同调计算任意单纯复形的所得同调。然后针对某些特殊类别的群，特别是元素中心化子为幂零的群，计算了具有此序层系数的团复形同调。
英文摘要
We study the clique complex $X_G$ of a finite group: a simplicial complex whose faces are in one-to-one correspondence with the elements of the group, and arise via a decomposition of group elements into $p$-elements for primes $p$. The $1$-skeleton of this complex then has the classical Gruenberg-Kegel graph $Γ_G$ as a quotient and a finite simple group is determined up to isomorphism by its clique complex. The clique complex can also be endowed with a combinatorial sheaf --- called the order sheaf --- and the resulting homology can be computed for an arbitrary simplicial complex in terms of the homology of the links of vertices. The homology of the clique complex with coefficients in this order sheaf is then computed for some special classes of groups, in particular those for which the centralizers of elements are nilpotent.
