Confined Subgroups of Simple Locally Finite Groups and Ideals of their Group Rings

We are concerned in this paper with the ideal structure of grouprings of infinite simple locally finite groups over fields ofcharacteristic zero, and its relation with certain subgroupsof the groups, called confined subgroups. The systematic studyof the ideals in these group rings was initiated by the secondauthor in15], although some results had been obtained previously(see 3, 1]). Let G be an infinite simple locally finite groupand K a field of characteristic zero. It is expected that inmost cases, the group ring KG will have the smallest possiblenumber of ideals, namely three, (KG itself, {0} and the augmentationideal), and this has been verified in some cases. In some interestingcases, however, the situation is different, and there are moreideals. We mention in particular the infinite alternating groups3] and the stable special linear groups 9], in which the ideallattice has been completely determined. The second author hasconjectured that the presence of ideals in KG, other than thethree unavoidable ones, is synonymous with the presence in thegroup of proper confined subgroups. Here a subgroup H of a locallyfinite group G is called confined, if there exists a finitesubgroup F of G such that H^gF1 for all gG. This amounts to sayingthat F has no regular orbit in the permutation representationof G on the cosets of H.