Cohomological periodicities of crystallographic groups

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• 作者：Graham Ellis ^{graham.ellis@nuigalway.ie" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cohomology of groups ; Crystallographic groups ; Finite p-groups
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：445
• 期：Complete
• 页码：537-544
• 全文大小：264 K

文摘

We observe that an n-dimensional crystallographic group G has periodic cohomology in degrees greater than n   if it contains a torsion free finite index normal subgroup S⊴G$S⊴G$ whose quotient G/S$G/S$ has periodic cohomology. We then consider a different type of periodicity. Namely, we provide hypotheses on a crystallographic group G   that imply isomorphisms H_i(G/γ_cT,F)≅H_i(G/γ_c+dT,F)$H_i(G/{\gamma }_{c}T,\mathbb{F})\cong H_i(G/{\gamma }_{c+d}T,\mathbb{F})$ for F$\mathbb{F}$ the field of p   elements and γ_cT${\gamma }_{c}T$ a term in the relative lower central series of the translation subgroup T≤G$T\le G$. The latter periodicity provides a means of calculating the mod-p homology of certain infinite families of finite p-groups using a finite (machine) computation.