transversals / lifts / sifts

Definition 1.

Given a group G and a subgroupMathworldPlanetmathPlanetmath H of G, a transversal of H in G is a subset TG such that for every gG there exists a unique tT such that Hg=Ht.

Typically one insists 1T so that the coset H is described uniquely by H1. However no standard terminology has emerged for transversals of this sort.

An alternative definition for a transversal is to use functions and homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath in a method more conducive to a categorical setting. Here one replaces the notion of a transversal as a subset of G and instead treats it as a certain type of map T:G/HG. Since H is generally not normal in G, G/H simply means the set of cosets, and T is therefore a function not a homomorphism. We only require that T satisfy the following property: Given the canonical projection map π:GG/H given by gHg (this is generally not a homomorphism either, and so both π and T are simply functions between sets) then πT=1G/H. It follows immediately that the image of T in G is a transversal in the original sense of the term.

Remark 2.

Because it is customary in group theory to write actions to the right of elements many times it is preferable to write Tπ=1G/H to match the right side notation.

When H is a normal subgroupMathworldPlanetmath of G our terminology adjusts from transversals to lifts.

Definition 3.

Given a group G and a homomorphism π:GQ, a lift of Q to G is a function f:QG such that πf=1Q.

It follows that π must be an epimorphismMathworldPlanetmathPlanetmath if it has a lift. Once again it is nearly always requested that f(1)=1 but this restrictionPlanetmathPlanetmathPlanetmath is generally not part of the definition.

Because both lifts and transversals are injectivePlanetmathPlanetmath mappings it is common to use the word lift/transversal for the image and the map with the context of the use providing any necessary clarification.

Definition 4.

Given a group G and a homomorphism π:GQ, a splitting map of Q to G is a homomorphism f:QG such that πf=1Q.

So we see a gradual progression in the definitions: We always have a group G and a set Q, and the maps π:GQ, f:QG satisfying


It follows, f is injective and π is surjectivePlanetmathPlanetmath.

  • f is a transversal if Q=G/H for some subgroup H. Here π and f are simply functions.

  • f is a lift if Q is a group. Here π is a homomorphism and f a function.

  • f is a splitting map if Q is group and both π and f are homomorphisms.

Finally we arrive at a stronger requirement for transversals and lifts which makes greater use of the group structureMathworldPlanetmath involved.

Definition 5.

Given a group G=S, there is a natural map π:F(S)G from the free groupMathworldPlanetmath on S onto G. A lift is a map l:GF(S) such that πl=1G. Furthermore a sift is a lift s:GF(S) with the added condition that sg=g for all gS.

Although a general sift is no more than a map that writes the elements of G as reduced words in S, in many cases the sifts have the added property of providing the words in a canonical form. This occurs when G=T0Tn-1 where Ti is a transversal of Gi/Gi+1. In such a case every element in G has a unique decomposition as a word t0t1tn-1 for unique tiTi.

Title transversals / lifts / sifts
Canonical name TransversalsLiftsSifts
Date of creation 2013-03-22 15:53:52
Last modified on 2013-03-22 15:53:52
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 11
Author Algeboy (12884)
Entry type Definition
Classification msc 20K27
Related topic SchreiersLemma
Related topic ExampleOfSchreiersLemma
Defines transversal
Defines lift
Defines sift