[GAP Forum] Test whether two groups have exactly the same elements with IsEqualSet.

[Hongyi Zhao]

Hi GAP team,

I try to test whether two groups have exactly the same elements with
IsEqualSet as follows:

f2 := FreeGroup("P", "Q");;
# assign variables
P:=f2.1;; Q:=f2.2;;
G4_2:= f2/[ [ P , P^-1 ], [ Q , Q^-1 ], [ Q *P, P *Q ] ];
g4_2_1:= f2 / [ P^2, Q^2, Comm( P, Q ) ];
g4_2_2:= f2 / [ P^2, Q^2, Comm( P, Q ) ];
Elements(G4_2); Elements(g4_2_1); Elements(g4_2_2);

IsEqualSet(Elements(G4_2), Elements(g4_2_1));
IsEqualSet(Elements(g4_2_1), Elements(g4_2_2));


But I obtained the following results:

gap> f2 := FreeGroup("P", "Q");;
gap> # assign variables
gap> P:=f2.1;; Q:=f2.2;;
gap> G4_2:= f2/[ [ P , P^-1 ], [ Q , Q^-1 ], [ Q *P, P *Q ] ];
<fp group on the generators [ P, Q ]>
gap> g4_2_1:= f2 / [ P^2, Q^2, Comm( P, Q ) ];
<fp group on the generators [ P, Q ]>
gap> g4_2_2:= f2 / [ P^2, Q^2, Comm( P, Q ) ];
<fp group on the generators [ P, Q ]>
gap> Elements(G4_2); Elements(g4_2_1); Elements(g4_2_2);
_2_1)); IsEqualSet(Elements(g4_2_1), Elements(g4_2_2));
[ <identity ...>, P, Q, P*Q ]
[ <identity ...>, P, Q, P*Q ]
[ <identity ...>, P, Q, P*Q ]
gap>
gap> IsEqualSet(Elements(G4_2), Elements(g4_2_1));
IsEqualSet(Elements(g4_2_1), Elements(g4_2_2));
false
false
gap>

As you can see, all three groups have identical elements. Why does the
IsEqualSet command still return false?

Regards
-- 
Assoc. Prof. Hongsheng Zhao <hongyi.zhao at gmail.com>
Theory and Simulation of Materials
Hebei Vocational University of Technology and Engineering
No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province