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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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gap> START_TEST("perm.tst");
gap> () = ();
true
gap> () = (1,2,3);
false
gap> (1,2,3) = ();
false
gap> (1,2,3) = (1,2,3);
true
gap> PermList([1,2,3]) = ();
true
gap> () = PermList([1,2,3]);
true
gap> (1,2) = PermList([2,1,3]);
true
gap> PermList([2,1,3]) = (1,2);
true
gap> checklens := Concatenation([1..20], [2^15-10..2^15+10],
>                               [2^16-10..2^16+10], [2^17-10..2^17+10]);;
gap> ForAll(checklens, n -> PermList(Concatenation([1..n-1], [n+1,n])) =
>                           PermList(Concatenation([1..n-1],[n+1,n,n+2])));
true
gap> ForAll(checklens, n -> not(
>  PermList(Concatenation([1..n-1], [n+1,n,n+2,n+4,n+3])) =
>  PermList(Concatenation([1..n-1], [n+1,n+2,n,n+4,n+3]))));
true
gap> STOP_TEST("perm.tst", 1840000);