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.
@article{urn:eudml:doc:41153,
title = {The p-period of an infinite group.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {241-250},
zbl = {0804.20041},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41153}
}
Yining, Xia. The p-period of an infinite group.. Publicacions Matemàtiques, Tome 36 (1992) pp. 241-250. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41153/