Numerical invariants for modular group algebras of Abelian groups. (Q2504973)

This work is a continuation of the author's papers [in Ric. Mat. 50, No. 2, 323-336 (2001; Zbl 1013.20004) and Rend. Circ. Mat. Palermo, II. Ser. 51, No. 3, 391-402 (2002; Zbl 1097.20504)] dealing with numerical invariants of commutative modular group algebras of prime characteristic \(p\). In this paper, certain invariants are provided for the pair \((FG,FA)\), where \(F\) is a field of characteristic \(p\) and \(A\) is a subgroup of the Abelian group \(G\). These include the dimensions of the vector spaces \(G^{p^\alpha}[p]A[p]/G^{p^{\alpha+1}}[p]A[p]\) and \((AG^{p^\alpha})[p]/(AG^{p^{\alpha+1}})[p]\) over the prime field with \(\alpha\) an arbitrary ordinal (generalizations of the classical Ulm-Kaplansky invariants), for instance. The proofs are based on arguments similar to those applied in the above-mentioned papers, such as establishing that the respective ideals, such as \(I(FG;G^{p^\alpha}[p]A[p])\) and \(I(FG;(AG^{p^\alpha})[p])\), are isomorphism invariants.