Elements of finite order in \(V(ZA_ 4)\)
Let G be a finite group and V(\({\mathbb{Z}}G)\) be the units of augmentation one in the integral group ring \({\mathbb{Z}}G\). In earlier papers [J. Algebra 66, 534-543 (1980; Zbl 0447.16006), Proc. Am. Math. Soc. 99, 9-14 (1987; Zbl 0616.16004), Commun. Algebra 16, 1479-1505 (1988; Zbl 0649.16009)], the authors have investigated the problem of whether G has a torsion free normal complement in V(\({\mathbb{Z}}G)\) for \(G=S_ 3\), \(A_ 4\), and \(S_ 4\). In particular they have shown that \(A_ 4\) has a torsion free normal complement in \(V({\mathbb{Z}}A_ 4)\). In the present paper the authors show that all normal complements to \(A_ 4\) in \(V({\mathbb{Z}}A_ 4)\) are torsion free. This is accomplished by finding all of the conjugate classes of elements of finite order in \(V({\mathbb{Z}}A_ 4)\) and then showing that a subgroup containing any such class must also contain an element of order 2 in \(A_ 4\).
scientific article; zbMATH DE number 4108931