On the normal complement problem of finite group algebra

Let G be a group, F be a finite field with a positive characteristic p > 0, then FG is the group algebra of a group G over F with a positive characteristic p > 0. In this paper, the normal complement problem on semisimple group algebra FG is examined. The normal complement problem of groups of order up to 16 in their corresponding unit groups U(FG) has already been discussed. We have proved that up to isomorphism all the three non-abelian groups of order 18 and up to isomorphism all three non-abelian groups of order 20 do not have normal complement in their corresponding unit groups U(FG).