Semigroup identities on units of group algebras
This paper is concerned with the following interesting question: for a group algebra \(k[G]\), how does it restrict the group \(G\) and \(k[G]\) itself if the group of units \({\mathcal U}(k[G])\) is group-theoretically ``somehow'' restricted. A semigroup identity is an equation \(u = v\) with \(u\) and \(v\) being some (distinct) words in a free semigroup. Due to \textit{A. Malcev}'s argument [Ivanov. Gos. Ped. Inst., Uchebn. Zap. Nauk 4, 107- 111 (1953; Zbl 0087.255)], a group satisfies a semigroup identity if it contains a normal nilpotent subgroup with quotient group being torsion of bounded exponent (and so, it includes the case of a nilpotent-by-finite group). The authors deal here with the question ``when does \({\mathcal U}(k[G])\) satisfy a semigroup identity?'' They prove (Th. 6) that for a torsion-generated group \(G\) and \({\mathcal U}(k[G])\) satisfying a semigroup identity, if \(\text{char }k = 0\), then the group \(G\) is Abelian, and if \(\text{char }k = p > 0\), then the quotient group \(G/{\mathcal Z}(G)\) is a \(p\)-group of bounded exponent, here by \({\mathcal Z}(G)\) is denoted the center of \(G\). For the case of a torsion group \(G\) and an infinite field \(k\) with \(\text{char }k = p\), the authors prove (see Th. 7) that the condition for \({\mathcal U}(k[G])\) to satisfy a semigroup identity is equivalent to \(G\) being nilpotent with its FC-subgroup \(\Delta(G)\) being such that both \(G/\Delta\) and \([\Delta,\Delta]\) are finite \(p\)-groups. Earlier, \textit{D. Warhurst} [Thesis, Manchester (1981)] had investigated a similar question for a \(p\)-group \(G\) with \({\mathcal U}(k[G])\) satisfying some group identities. Among other results of this interesting and clear-written paper note (Th. 12): for a finitely generated \(p\)-group \(G\) and an uncountable field \(k\) of characteristic \(p>0\), either there exists an integer \(n>0\) such that \(x^{p^ n}\) is a central polynomial for \(k[G]\) or \({\mathcal U}(k[G])\) contains a noncommutative free subgroup.
scientific article; zbMATH DE number 62246