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Partial representations and partial group algebras. (English) Zbl 0954.20004
Let \(G\) be a finite group and \(K\) a field. The partial group \(K\)-algebra \(K_{par}(G)\) is the universal \(K\)-algebra with unit \(1\) generated by the symbols \([g]\), \(g\in G\), with relations \([e]=1\), \([s^{-1}][s][t]=[s^{-1}][st]\), \([s][t][t^{-1}]=[st][t^{-1}]\). The representations of \(K_{par}(G)\) are in one-to-one correspondence with the so called partial representations of \(G\).
The authors investigate the structure of \(K_{par}(G)\), and they prove that if \(G_1\) and \(G_2\) are finite Abelian groups and the characteristic does not divide the order of \(G_1\), then \(K_{par}(G_1)\) is isomorphic to \(K_{par}(G_2)\) if and only if \(G_1\) is isomorphic to \(G_2\).

MSC:
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
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