The strange isomorphic mapping shown in GAP: [ P, Q ] -> [ <[ [ 1, 1 ] ]|P>, <[ [ 2, 1 ] ]|Q> ].

Question

I tested the following code snippet in GAP:

gap> f:=FreeGroup("P" ,"Q");;
gap> G8_5:=f/ParseRelators(f, "P^4 = Q^4 = 1, Q*P = P^3*Q, Q^2 = P^2");;
gap> AssignGeneratorVariables(G8_5);
#I  Global variable `P' is already defined and will be overwritten
#I  Global variable `Q' is already defined and will be overwritten
#I  Assigned the global variables [ P, Q ]
gap> g1:=Group( Elements(G8_5) );
Group([ P, P^3, Q, P^2*Q, P^2, P*Q, P^3*Q ])
gap> IsomorphismGroups(G8_5,g1);
#I  Forcing finiteness test
[ P, Q ] -> [ <[ [ 1, 1 ] ]|P>, <[ [ 2, 1 ] ]|Q> ]

As you can see, it gives the description of the isomorphic mapping represented by the following relationship:

[ P, Q ] -> [ <[ [ 1, 1 ] ]|P>, <[ [ 2, 1 ] ]|Q> ]

But the form of image elements looks strange, and I failed to figure out the meaning of them. Any hints will be highly appreciated.

Regards, HZ

Answer 1

The notation < [[a,b],[...]]| elm > describes elements that are (potentially) stored as a product in generators (to save on memory). The actual element is what is given between | and >, i.e. $P$ and $Q$ in your case, and these elements will compare as being equal to $P$ and $Q$.