[GAP Forum] The function DirectProduct

[mike newman]

On Wed, 14 Dec 2011, Luy®n Lê Vån wrote:
>
> When I ran the below code lines:
> gap >S:=SymmetricGroup(3);;
> gap >G:=DirectProduct(S);;
> gap >G=S;
>        true
>
> gap >C:=CyclicGroup(3);;
> gap >G:=DirectProduct(C);;
> gap >G=C;
>        false


The issue is that "equal" and "isomorphic" are not the same thing for 
groups (among other structures).  In fact in GAP there is also the 
notion of "identical", but that doesn't apply to your example.
So identical==>equal==>isomorphic, but the converses are false.

For your example:

gap> S:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> G:=DirectProduct(S);
Group([ (1,2,3), (1,2) ])
gap> G=S;
true
gap> IsomorphismGroups(G,S);
[ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ]

gap> C:=CyclicGroup(3);
<pc group of size 3 with 1 generators>
gap> G:=DirectProduct(C);
<pc group of size 3 with 1 generators>
gap> G=C;
false
gap> IsomorphismGroups(G,C);
[ f1 ] -> [ f1 ]

gap> S=C;
false
gap> IsomorphismGroups(S,C);
fail


--mike