# The p-period of an infinite group.

Publicacions Matemàtiques (1992)

- Volume: 36, Issue: 1, page 241-250
- ISSN: 0214-1493

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topYining, Xia. "The p-period of an infinite group.." Publicacions Matemàtiques 36.1 (1992): 241-250. <http://eudml.org/doc/41153>.

@article{Yining1992,

abstract = {For Γ a group of finite virtual cohomological dimension and a prime p, the p-period of Γ is defined to be the least positive integer d such that Farrell cohomology groups Hi(Γ; M) and Hi+d(Γ; M) have naturally isomorphic ZΓ modules M.We generalize a result of Swan on the p-period of a finite p-periodic group to a p-periodic infinite group, i.e., we prove that the p-period of a p-periodic group Γ of finite vcd is 2LCM(|N(〈x〉) / C(〈x〉)|) if the Γ has a finite quotient whose a p-Sylow subgroup is elementary abelian or cyclic, and the kernel is torsion free, where N(-) and C(-) denote normalizer and centralizer, 〈x〉 ranges over all conjugacy classes of Z/p subgroups. We apply this result to the computation of the p-period of a p-periodic mapping class group. Also, we give an example to illustrate this formula is false without our assumption.},

author = {Yining, Xia},

journal = {Publicacions Matemàtiques},

keywords = {Grupos infinitos; Grupos periódicos; Cohomología; Periodicidad; virtual cohomological dimension; subgroup of finite index; Farrell cohomology; Tate cohomology; -period; -primary components; - periodic group; mapping class groups},

language = {eng},

number = {1},

pages = {241-250},

title = {The p-period of an infinite group.},

url = {http://eudml.org/doc/41153},

volume = {36},

year = {1992},

}

TY - JOUR

AU - Yining, Xia

TI - The p-period of an infinite group.

JO - Publicacions Matemàtiques

PY - 1992

VL - 36

IS - 1

SP - 241

EP - 250

AB - For Γ a group of finite virtual cohomological dimension and a prime p, the p-period of Γ is defined to be the least positive integer d such that Farrell cohomology groups Hi(Γ; M) and Hi+d(Γ; M) have naturally isomorphic ZΓ modules M.We generalize a result of Swan on the p-period of a finite p-periodic group to a p-periodic infinite group, i.e., we prove that the p-period of a p-periodic group Γ of finite vcd is 2LCM(|N(〈x〉) / C(〈x〉)|) if the Γ has a finite quotient whose a p-Sylow subgroup is elementary abelian or cyclic, and the kernel is torsion free, where N(-) and C(-) denote normalizer and centralizer, 〈x〉 ranges over all conjugacy classes of Z/p subgroups. We apply this result to the computation of the p-period of a p-periodic mapping class group. Also, we give an example to illustrate this formula is false without our assumption.

LA - eng

KW - Grupos infinitos; Grupos periódicos; Cohomología; Periodicidad; virtual cohomological dimension; subgroup of finite index; Farrell cohomology; Tate cohomology; -period; -primary components; - periodic group; mapping class groups

UR - http://eudml.org/doc/41153

ER -

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