# How can you operate in a quotient of a group finitely presented?

Hello everyone, I have the following groups

G.<a,b> = FreeGroup()
H = G.quotient([a*b*a.inverse()*b.inverse()])


I would like SAGE to understand the product of to lateral clases in H? It is that possible?

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What kind of product should be constructed?

We have for instance:

sage: G.<a,b> = FreeGroup()
sage: H = G.quotient( [ a*b*a.inverse()*b.inverse() ] )
sage: H.is_abelian()
True
sage: h = H( a^3 * b^4 ) * H( a^-7 * b^2 )
sage: h
a^3*b^4*a^-7*b^2
sage: h == H(a^-4 * b^6)
True

sage: A.<s,t> = AbelianGroup( 2 )
sage: A
Multiplicative Abelian group isomorphic to Z x Z
sage: ( s^3 * t^4 ) * ( s^-7 * t^2 )
s^-4*t^6


The form a^3*b^4*a^-7*b^2 for the above product computed in H is a possible one. There is no simplification, since there is no "canonical form" to simplify in a general quotient. But the equality h == H(a^-4 * b^6) could be verified.

Is there any reason to use H and not A as above?

( 2017-05-10 13:50:18 -0600 )edit