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

asked 2017-05-04 19:56:13 +0100

anonymous user

Anonymous

updated 2017-05-08 00:05:39 +0100

tmonteil gravatar image

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?

Thanks in advance!

edit retag flag offensive close merge delete

Comments

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?

dan_fulea gravatar imagedan_fulea ( 2017-05-10 20:50:18 +0100 )edit