[GAP Forum] Van Kampen theoriem for the Klein bottle applied to fp groups

[Mbg Nimda]

Hello forum members,

I hope someone can point me out what is missing in the following:
I divide the Klein bottle into two Moebius strips that intersect on a
tubular
neighbourhood that is homotopically equivalent to S^1. The embedding of
this loop
is a^2 in one Moebius strip and b^2 in the other, where a and b are the
homotopy equivalent
loops of each Moebius strip respectively (the border of a Moebius strip
retracts to twice
the "middle circle" of the strip). So the Van Kampen theorem says that the
fundamental group
is the fp group with generators a and b and relator a^2*b^-2.
Now I looked up the funcamental group of the Klein bottle on Google and
found that the litterature declares the
fundamental group to have the relator a*b *a^-1*b (due to how opposite
sides on a square are identified) .
So initially I thought I was wrong but when frolicking around with these
two relators I found the following:
a*b *a^-1*b = 1 implies a*b*a^-2a*b. So if I put u:=a*b and v :=a then the
last equation becomes
u*v^-2*u = 1 or u^2*v^-2, so the two presentations are equivalent.
I found this relationship totally by good luck, but I wonder if there could
have been a way
using Gap to establish an isomorphism between those two groups?

Regards,

Marc Bogaerts