an:05676092
Zbl 1201.20003
Lebel, Alain; Flannery, D. L.; Horadam, K. J.
Group algebra series and coboundary modules
EN
J. Pure Appl. Algebra 214, No. 7, 1291-1300 (2010).
00258948
2010
j
20C05 20J06 16S34
group algebras; Loewy series; socle series; cohomology; 2-cocycles; coboundaries; shift actions; modules; finite \(p\)-groups
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.
J??nos Kurdics (Ny??regyh??za)
Zbl 1043.20026