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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
Software:
Magma
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References:
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