# transversals / lifts / sifts

###### Definition 1.

Given a group $G$ and a subgroup $H$ of $G$, a transversal of $H$ in $G$ is a subset $T\subseteq G$ such that for every $g\in G$ there exists a unique $t\in T$ such that $Hg=Ht$.

Typically one insists $1\in T$ 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 homomorphisms 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 $T:G/H\rightarrow G$. 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 $\pi:G\rightarrow G/H$ given by $g\mapsto Hg$ (this is generally not a homomorphism either, and so both $\pi$ and $T$ are simply functions between sets) then $\pi T=1_{G/H}$. 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 $T\pi=1_{G/H}$ to match the right side notation.

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

###### Definition 3.

Given a group $G$ and a homomorphism $\pi:G\rightarrow Q$, a lift of $Q$ to $G$ is a function $f:Q\rightarrow G$ such that $\pi f=1_{Q}$.

It follows that $\pi$ must be an epimorphism if it has a lift. Once again it is nearly always requested that $f(1)=1$ but this restriction is generally not part of the definition.

Because both lifts and transversals are injective 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 $\pi:G\rightarrow Q$, a splitting map of $Q$ to $G$ is a homomorphism $f:Q\rightarrow G$ such that $\pi f=1_{Q}$.

So we see a gradual progression in the definitions: We always have a group $G$ and a set $Q$, and the maps $\pi:G\rightarrow Q$, $f:Q\rightarrow G$ satisfying

 $\pi f=1_{Q}.$

It follows, $f$ is injective and $\pi$ is surjective.

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

• $f$ is a lift if $Q$ is a group. Here $\pi$ is a homomorphism and $f$ a function.

• $f$ is a splitting map if $Q$ is group and both $\pi$ and $f$ are homomorphisms.

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

###### Definition 5.

Given a group $G=\langle S\rangle$, there is a natural map $\pi:F(S)\rightarrow G$ from the free group on $S$ onto $G$. A lift is a map $l:G\rightarrow F(S)$ such that $\pi l=1_{G}$. Furthermore a sift is a lift $s:G\rightarrow F(S)$ with the added condition that $sg=g$ for all $g\in S$.

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 $G=T_{0}\cdots T_{n-1}$ where $T_{i}$ is a transversal of $G^{i}/G^{i+1}$. In such a case every element in $G$ has a unique decomposition as a word $t_{0}t_{1}\cdots t_{n-1}$ for unique $t_{i}\in T_{i}$.

Title transversals / lifts / sifts TransversalsLiftsSifts 2013-03-22 15:53:52 2013-03-22 15:53:52 Algeboy (12884) Algeboy (12884) 11 Algeboy (12884) Definition msc 20K27 SchreiersLemma ExampleOfSchreiersLemma transversal lift sift