Isomorphism of modular group algebras of abelian groups with semi-complete $p$-primary components
Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 157-161
Printed June 1, 2007
Peter Danchev
Abstract : Let $G$ be a $p$-mixed abelian group with semi-complete torsion subgroup $G_t$ such that $G$ is splitting or is of torsion-free rank one, and let $R$ be a commutative unitary ring of prime characteristic $p$. It is proved that the group algebras $RG$ and $RH$ are $R$-isomorphic for any group $H$ if and only if $G$ and $H$ are isomorphic. This isomorphism relationship extends our earlier results in (Southeast Asian Bull. Math., 2002), (Proc. Amer. Math. Soc., 2002) and (Bull. Korean Math. Soc., 2005) as well as completely settles a problem posed by W. May in (Proc. Amer. Math. Soc., 1979).
Keywords : group algebras, isomorphisms, semi-complete groups, $p$-mixed splitting groups, $p$-mixed groups with torsion-free rank one
MSC numbers : 20C07, 16S34, 16U60, 20K10, 20K15, 20K20, 20K21
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