# Problem while multiplying under a set of relators

I am trying to multiply two elements of a group-ring.

``````gap> f := FreeGroup( "a", "b","c" );;

gap> g := f / ( f.1^2, f.2^3,f.3^4, f.1*f.2*f.3 );

<fp group on the generators ( a, b, c )>

gap> G := f / ( f.1^2, f.2^3,f.3^4, f.1*f.2*f.3 );

<fp group on the generators ( a, b, c )>

gap> F:=GF(5);

GF(5)

gap> FG:=GroupRing(F,G);

<algebra-with-one over GF(5), with 3 generators>

gap> i:=Embedding(G,FG);

<mapping: Group( ( a, b, c ) ) -> AlgebraWithOne( GF(5), ... ) >

gap> e:=One(FG);

(Z(5)^0)*<identity ...>

gap> List(G);

( <identity ...>, a, b, b^-1, c, c^-1, a*b^-1, a*c, b*a, b*c^-1, b^-1*c^-1, c*b^-1, c^2,
c^-1*a, a*b^-1*c^-1, a*c*b^-1, b*a*b^-1, b*a*c, b^-1*c^-1*a, c*b^-1*c^-1, c^-1*a*c,
a*b^-1*c^-1*a, b*a*b^-1*c^-1, b^-1*c^-1*a*c )

gap> a:=G.1;

a

gap> B:=G.2;

b

gap> b:=G.2;

b

gap> c:=G.3;

c

gap> U1:=e+(e+a)*b*(e-a);

(Z(5)^0)*<identity ...>+(Z(5)^2)*a*b*a+(Z(5)^0)*b+(Z(5)^2)*b*a+(Z(5)^0)*a*b

gap> U2:=e+(e-a)*b*(e+a);

(Z(5)^0)*<identity ...>+(Z(5)^2)*a*b*a+(Z(5)^0)*b+(Z(5)^0)*b*a+(Z(5)^2)*a*b

gap> U1*U2;

(Z(5)^0)*<identity ...>+(Z(5))*(a*b*a)^2+(Z(5)^3)*(b*a)^2+(Z(5)^3)*(a*b)^2+(Z(5))*b*a^2*b*a+(
Z(5)^3)*a*b*a+(Z(5))*b+(Z(5))*b*a^2*b+(Z(5)^3)*b*a*b+(Z(5)^3)*(a*b)^2*a+(Z(5))*a*b^2
``````

Question
Why I am not getting simplified form of the product obtained in the last step ?

Since `a^2=e`, but it is showing like `(Z(5))*b*a^2*b`.