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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

Hongyi Zhao
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  • This is not a question about Mathematics. You're on the wrong forum. – jjagmath May 14 '22 at 10:40
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    I think you need to more specific and explain what is "strange". Is it strange that P maps to P and Q to Q? Or, as I may guess, you haven't seen <[ [ 1, 1 ] ]|P> in the output before, and you'd like to know what does it mean? Note that g1 is created as a subgroup of G8_5, is that would you intended to do? – Olexandr Konovalov May 14 '22 at 13:43
  • By "strange", I mean I can't figure out the meaning of <[ [ 1, 1 ] ]|P> and <[ [ 2, 1 ] ]|Q>, and what's the mapping action (mapping function) in detail. 2. ```gap> IsSubgroup(G8_5,g1);

    • true gap> IsSubgroup(g1,G8_5); true```, so they're subgroups of each other. Why do you only say g1 is created as a subgroup of G8_5?

    – Hongyi Zhao May 14 '22 at 13:57
  • I am unsure what you're trying to achieve. You're generating g1 by the whole list of Elements(G8_5). Clearly g1 it's a subgroup of G8_5 then, and since each generator of G8_5 is also a generator of g1, then g1 coincides with G8_5. So they are isomorphic, and a cheaper way to establish that fact in this case is to ask g1=G8_5 (unless you want an explicit isomorphism). – Olexandr Konovalov May 14 '22 at 14:42
  • My purpose: Verify that two groups are the same by using different forms of group elements based on the same generators set. The example here uses the same group, so it's not easy to explain. 2. I still want to know the implication of the symbol: <[ [ 1, 1 ] ]|P>.
    • – Hongyi Zhao May 14 '22 at 15:00