Groupe

F := FreeGroup( "a", "b");
<free group on the generators [ a, b ]>
Size(F);

infinity

G := F/[F.1*F.2/F.1/F.2, F.1^3, F.2^4];
<fp group on the generators [ a, b ]>
Size(G);
12
GeneratorsOfGroup(G);
[a,b]
a := g.1; 
b := g.2;
a
b
AssignGeneratorVariables(G);
//    Idem
a*a*a*a = a 
true
a in G;
a in F; 
true
false
a^3 = One(G);
a^3 = One(F) ;
true
false
(a*b)^50;
(a*b)^50
SetReducedMultiplication(G);
 
(a*b)^50;
a^-1*b^2
IsSubgroupFpGroup(G);
true
IsGroupOfFamily(G);
true
IsFpGroup(G);
//    = IsSubgroupFpGroup(G) and IsGroupOfFamily(G)

 

G := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
H := CyclicGroup(5);
<pc group of size 5 with 1 generators>
IsFpGroup(G);
false
AsList(H);
[ <identity> of ..., f1, f1^2, f1^3, f1^4 ]
L := FreeGroup("a","b");
AssignGeneratorVariables(L);
G := FactorGroupFpGroupByRels(L, [a*b^3, b*a^3]);
Size(G);
AsList(G);
IsAbelian(G);
<free group on the generators [ a, b ]>
#I Assigned the global variables [ a, b ]
<fp group on the generators [ a, b ]>
8
[ <identity ...>, a, a^-1, b, b^-1, a^2, a*b, a*b^-1 ]
true