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Lifts Transversals
and a subgroup
of 
, a transversal of 
in 
is a subset
such that for every 
there exists a unique
such that 
.
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.
to match the right side
notation.
When 
is a normal subgroup of 
our terminology adjusts from transversals to lifts.
and a homomorphism
, a lift of 
to 
is a function
such that 
.
It follows that 
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.
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
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.
, 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
.