The semiprimitivity problem for group algebras of locally finite groups

LetK[G] be the group algebra of a locally finite groupG over a fieldK of characteristicp>0. IfG has a locally subnormal subgroup of order divisible byp, then it is easy to see that the Jacobson radical ?K[G] is not zero. Here, we come close to a complete converse by showing that ifG has no nonidentity locally subnormal subgroups, thenK[G] is semiprimitive. The proof of this theorem uses the much earlier semiprimitivity results on locally finite, locallyp-solvable groups, and the more recent results on locally finite, infinite simple groups. In addition, it uses the beautiful properties of finitary permutation groups.