CONFINED SUBGROUPS OF SIMPLE LOCALLY FINITE GROUPS AND IDEALS OF THEIR GROUP RINGS.

We are concerned in this paper with the ideal structure of group rings of infinite simple locally finite groups over fields of characteristic zero, and its relation with certain subgroups of the groups, called confined subgroups. The systematic study of the ideals in these group rings was initiated by the second author in [15], although some results had been obtained previously (see [3, 1]). Let G be an infinite simple locally finite group and K a field of characteristic zero. It is expected that in most cases, the group ring KG will have the smallest possible number of ideals, namely three, (KG itself, {0} and the augmentation ideal), and this has been verified in some cases. In some interesting cases, however, the situation is different, and there are more ideals. We mention in particular the infinite alternating groups [3] and the stable special linear groups [9], in which the ideal lattice has been completely determined. The second author has conjectured that the presence of ideals in KG, other than the three unavoidable ones, is synonymous with the presence in the group of proper confined subgroups. Here a subgroup H of a locally finite group G is called confined, if there exists a finite subgroup F of G such that Hg)F`1 for all gG. This amounts to saying that F has no regular orbit in the permutation representation of G on the cosets of H. [ABSTRACT FROM AUTHOR]