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Associated primes in group cohomology and a strange result
Abstract
In 2010. Okuyama proved the following. Let G be a finite group and k an algebraically closed field of characteristic p dividing the order of G. (1) Let frak p be an associated prime of H*(G,k) of dim=s. Then there exists an elementary abelian p-subgroup E⊆ G of rank s with p=pE and depthH*(CGE,k)=s. (2) Let E be an elementary abelian p-subgroup of G of rank s and suppose depthH*(CGE,k)=s. Then pE is an associated prime of H*(G,k). It is shown this result is true for all fields of characteristic p. We also show the following result: If E is an elementary abelian p-subgroup of G and depth H*(G,k)=rank E, then up to conjugation E contains the maximal elementary abelian subgroup contained in the center of a Sylow p-subgroup of G.
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BibTeX
@misc{schafer2026associated,
title = {{Associated primes in group cohomology and a strange result}},
author = {James A. Schafer},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03569},
eprint = {2611.03569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03569},
}
RIS
TY - UNPB
TI - Associated primes in group cohomology and a strange result
AU - James A. Schafer
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03569
UR - https://arxiv.org/abs/2611.03569
AB - In 2010. Okuyama proved the following. Let G be a finite group and k an algebraically closed field of characteristic p dividing the order of G. (1) Let frak p be an associated prime of H^*(G,k) of dim=s. Then there exists an elementary abelian p-subgroup E⊆ G of rank s with p=p_E and depthH^*(C_GE,k)=s. (2) Let E be an elementary abelian p-subgroup of G of rank s and suppose depthH^*(C_GE,k)=s. Then p_E is an associated prime of H^*(G,k). It is shown this result is true for all fields of characteristic p. We also show the following result: If E is an elementary abelian p-subgroup of G and depth H^*(G,k)=rank E, then up to conjugation E contains the maximal elementary abelian subgroup contained in the center of a Sylow p-subgroup of G.
ER -