This works:

gap> G:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> R:=GroupRing(Integers,G);
<free left module over Integers, and ring-with-one, with 2 generators>
gap> m:=Embedding(G,R);
<mapping: SymmetricGroup( [ 1 .. 4 ] ) -> FLMLORWithOne( Integers, ... ) >
gap> ( ()^m + 2*(1,3)^m + 3*(3,2,4)^m ) * ( 4*(2,1,4)^m + 2*(1,3,2,4)^m );
(4)*(1,2,4)+(14)*(1,3,4,2)+(2)*(1,3,2,4)+(4)*(1,4,2)+(12)*(1,4,3)
gap> gens:=[(1,2)^m, (1,3)^m, (1,4)^m];
[ (1)*(1,2), (1)*(1,3), (1)*(1,4) ]

The last line gives you your generating set that you asked for, but you
don't actually need it.

On Sun, 30 Dec 2001, Rex L Agacy wrote:

Dear GAP-Forum members,
As a complete beginner can anyone say how to set up the algebraic structure
so as to perform multiplication of elements in the group algebra of S_n, eg
for S_4 (with generators specified as say (1,2),(1,3),(1,4) - but how?), I'd
like to calculate: ( () + 2*(1,3) + 3*(3,2,4) ) * ( 4*(2,1,4) +
2*(1,3,2,4) ) where () is the id permutation and the others are cycles -
coefficients are integers?
Could the answers be added to the Tutorial manual?
Much obliged.
Thanks
RL Agacy