PlanetMath: transversals / lifts / sifts

(more info)

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transversals / lifts / sifts

(Definition)

Definition 1 Given a group and a subgroup of , a transversal of in is a subset $T\subseteq G$ such that for every $g\in G$ there exists a unique $t\in T$ such that .

Typically one insists $1\in T$ 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 $T:G/H\rightarrow G$ . 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 $\pi:G\rightarrow G/H$ given by $g\mapsto Hg$ (this is generally not a homomorphism either, and so both $\pi$ and are simply functions between sets) then $\pi T=1_{G/H}$ . It follows immediately that the image of in 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 is a normal subgroup of our terminology adjusts from transversals to lifts.

Definition 3 Given a group and a homomorphism $\pi:G\rightarrow Q$ , a lift of to 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 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 and a homomorphism $\pi:G\rightarrow Q$ , a splitting map of to 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 and a set , and the maps $\pi:G\rightarrow Q$ , $f:Q\rightarrow G$ satisfying

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

It follows,

is injective and $\pi$ is surjective.

is a transversal if for some subgroup . Here $\pi$ and are simply functions.
is a lift if is a group. Here $\pi$ is a homomorphism and a function.
is a splitting map if is group and both $\pi$ and 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 onto . 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 for all $g\in S$ .

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 $G=T_0\cdots T_{n-1}$ where is a transversal of $G^i/G^{i+1}$ . In such a case every element in has a unique decomposition as a word $t_0t_1\cdots t_{n-1}$ for unique $t_i\in T_i$ .