transversals / lifts / sifts
Typically one insists so that the coset is described uniquely by . However no standard terminology has emerged for transversals of this sort.
An alternative definition for a transversal is to use functions and homomorphisms in a method more conducive to a categorical setting. Here one replaces the notion of a transversal as a subset of and instead treats it as a certain type of map . Since is generally not normal in , simply means the set of cosets, and is therefore a function not a homomorphism. We only require that satisfy the following property: Given the canonical projection map given by (this is generally not a homomorphism either, and so both and are simply functions between sets) then . It follows immediately that the image of in is a transversal in the original sense of the term.
When is a normal subgroup of our terminology adjusts from transversals to lifts.
Given a group and a homomorphism , a lift of to is a function such that .
Given a group and a homomorphism , a splitting map of to is a homomorphism such that .
So we see a gradual progression in the definitions: We always have a group and a set , and the maps , satisfying
It follows, is injective and is surjective.
is a transversal if for some subgroup . Here and are simply functions.
is a lift if is a group. Here is a homomorphism and a function.
is a splitting map if is group and both and are homomorphisms.
Finally we arrive at a stronger requirement for transversals and lifts which makes greater use of the group structure involved.
Given a group , there is a natural map from the free group on onto . A lift is a map such that . Furthermore a sift is a lift with the added condition that for all .
Although a general sift is no more than a map that writes the elements of as reduced words in , in many cases the sifts have the added property of providing the words in a canonical form. This occurs when where is a transversal of . In such a case every element in has a unique decomposition as a word for unique .
|Title||transversals / lifts / sifts|
|Date of creation||2013-03-22 15:53:52|
|Last modified on||2013-03-22 15:53:52|
|Last modified by||Algeboy (12884)|