Prove that all the groups below belong to the same isomorphism class.
Notation means group generated by the set U. Having the inclusion in a group (or semi-group) X it should be clear what is the operation on the set.
is the set of all possible functions .
is the field of rational functions of the variable x over field .
is the group of all invertible matrices with elements in field (ring) .
denotes the presentation of the group: is a set of (abstract) generators, elements of together with their conjugates generate a subgroup that is annihilated (kernel of a homomorphism).