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Group algebra series and coboundary modules. (English) Zbl 1201.20003
The shift action on the 2-cocycle group $$Z^2(G,C)$$ of a finite group $$G$$ with coefficients in a finitely generated Abelian group $$C$$, introduced by the third author [J. Pure Appl. Algebra 188, No. 1-3, 127-143 (2004; Zbl 1043.20026)], has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action.
In this article, the authors study the shift orbit structure of the coboundary subgroup $$B^2(G,C)$$ of $$Z^2(G,C)$$. The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over $$G$$. They prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if $$C$$ is an elementary Abelian $$p$$-group, then almost all shift orbits in $$B^2(G,C)$$ are maximal-sized for large enough finite $$p$$-groups $$G$$ of certain classes.

##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20J06 Cohomology of groups 16S34 Group rings
Magma
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##### References:
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