THE SEMIPRIMITIVITY PROBLEM FOR GROUP-ALGEBRAS OF LOCALLY FINITE-GROUPS

Let K[G] be the group algebra of a locally finite group G over a held K of characteristic p > 0. If G has a locally subnormal subgroup of or der divisible by p, then it is easy to see that the Jacobson radical J K[G] is not zero. Here, we come close to a complete converse by showin g that if G has no nonidentity locally subnormal subgroups, then K[G] is semiprimitive. The proof of this theorem uses the much earlier semi primitivity results on locally finite, locally p-solvable groups, and the more recent results on locally finite, infinite simple groups. In addition, it uses the beautiful properties of finitary permutation gro ups.