Units in group rings: Splittings and the isomorphism problem (Q1073142)

In 1975, \textit{R. K. Dennis} posed the question of when the natural injection \(G\to V(RG)\) of the theory of group rings is the section map of a splitting, so that \(V(RG)\) is of the form \(N\leftthreetimes G\) [see Ring theory II, Proc. 2nd Okla. Conf. 1975, 103-130 (1977; Zbl 0357.16003)]. For the most important case \(R={\mathbb{Z}}\), several results, especially those of \textit{G. H. Cliff}, \textit{S. K. Sehgal} and \textit{A. R. Weiss} [see J. Algebra 73, 167-185 (1981; Zbl 0484.16004)] and of the authors [see J. Pure Appl. Algebra 27, 299-314 (1983; Zbl 0509.16006)] suggested that the answer is affirmative mainly for metabelian groups G of the form \(1\to A\to G\to G/A\to 1,\) where G/A is of odd order. The present paper strengthens this feeling by showing the following Theorem. If G is non-abelian, simple and not isomorphic to a PSL(n,q) \((n>1\), q a prime power), then \(G\to V({\mathbb{Z}}G)\) is the section of a splitting. They also extend the result of Cliff-Sehgal-Weiss to the modular case (cf. Theorem III in the paper). For this case, they suppose that A and G/A are abelian p-groups, A being elementary, and that \(R={\mathbb{Z}}/p{\mathbb{Z}}\). Also, in order to prove the existence of the splitting, they have to introduce another definition which produces a group which is different from V(RG) in this case. The proof of the theorem stated above, which is the main result in this paper, follows from an adapted version of the congruence subgroup theorem of \textit{H. Bass}, \textit{J. Milnor} and \textit{J. P. Serre} [see Inst. Haut. Etud. Sci., Publ. Math. 33, 59-137 (1967; Zbl 0174.052)]. The main observation is that, fixing a maximal order \(\Lambda\) in \({\mathbb{Q}}G\) containing \({\mathbb{Z}}G\), any splitting epimorphism \(V({\mathbb{Z}}G)\to G\) induces an epimorphism SL(\({\mathbb{Z}}G)\to G\) whose kernel contains a congruence subgroup of \(\Lambda\). Using this, the authors are able to show that, under the stated assumption, G has a PSL(n,q) as a quotient, and from this the theorem follows obviously. The paper treats also some consequences for the isomorphism problem and, in particular, gives a new result, valid for the modular case \(R={\mathbb{Z}}/p{\mathbb{Z}}\), which extends a classical result of \textit{I. B. S. Passi} and \textit{S. K. Sehgal} [see Math. Z. 129, 65-73 (1972; Zbl 0234.20003)].