Units in the modular group algebra of a finite Abelian p-group. (English) Zbl 0543.20008

The author presents two minimal generating sets for the group V of normalized units in \({\mathbb{F}}_ pA\), A an abelian p-group. One is given as follows: Let A have ”basis” \(x_ i\) of order \(q_ i\), \(1\leq i\leq d\) and \(let\)
D\(=\{\delta =(\delta_ 1,...,\delta_ d): \delta_ i\in {\mathbb{N}}_ 0\), \(0\leq \delta_ i<q_ i\), \(p| \delta_ j\) for some \(j\}\).
For \(\delta \in D\) put \(P(\delta)=\prod_{1\leq j\leq d}(x_ j- 1)^{\delta_ i}\). Then \(\{1+P(\delta): \delta \in D\}\) is a basis for V. A similar basis is given in terms of the socle of G. Also the invariants and the rank of V are determined. Finally the author proves the interesting result: Let A be a finite abelian p-group. Then a subgroup U of \(V({\mathbb{F}}_ pA)\) which is linearly independent as a subset of \({\mathbb{F}}_ pA\) is isomorphic to a subgroup of A. The analogous result is also true for the non-abelian group of order \(p^ 3\), but not for the quaternion group of order 16.
Reviewer: K.W.Roggenkamp


20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
Full Text: DOI


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