发表机构
Universidad Veracruzana（韦拉克鲁斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Frattini拓扑约化将有限群的非生成复形化为初等阿贝尔向量空间的非张成复形，从而确定其同构、自同构群、同调及Betti数，并给出完整Cohen-Macaulay分类。
AI 中文摘要
对于有限群\\(G\\)，设\\(N(G)\\)为不生成\\(G\\)的子集所构成的单纯复形。我们将一般的Frattini拓扑约化与有限\\(p\\)-群特有的结构分离开来。对于每个有限非循环群，非锥Frattini核同伦等价于\\(G/\Phi(G)\\)的子群格的适当部分的有序复形。对于有限\\(p\\)-群，该商是初等阿贝尔向量空间，核成为\\(\mathrm{PG}(r-1,p)\\)的一致平行扩张的非张成复形，其中\\(r=d(G)\\)且\\(q=|\Phi(G)|\\)。我们利用该几何确定精确的单纯同构数据和完全单纯自同构群，将顶部同调与Steinberg模的适当限制等变地等同起来，并推导出模论推论。我们还给出了每个多重分次Betti数的显式支撑公式，并将其与\\(\mathbb Z\\)-分次Betti数的封闭单和公式进行比较。已知的拟阵和建筑理论输入与群特有的结论分开陈述。所得框架还产生了同伦类型、深度、正则性、射影维数、面枚举以及完整的Cohen-Macaulay分类。
英文摘要
For a finite group \(G\), let \(N(G)\) be the simplicial complex of subsets that do not generate \(G\). We separate a general Frattini-topological reduction from the structure special to finite \(p\)-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of \(G/Φ(G)\). For a finite \(p\)-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of \(\mathrm{PG}(r-1,p)\), where \(r=d(G)\) and \(q=|Φ(G)|\). We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the \(\mathbb Z\)-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen--Macaulay classification.
Comments14 pages. Preprint
