transversals / lifts / sifts

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

⊆

G

such that for every

g

∈

G

there exists a unique

t

∈

T

such that

H

⁢

g

=

H

⁢

t

.

Typically one insists

1

∈

T

so that the coset

H

is described uniquely
by

H

⁢

1

. 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

→

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

π

:

G

→

G

/

H

given by

g

↦

H

⁢

g

(this is generally not a homomorphism either, and so both

π

and

T

are simply functions between sets) then

π

⁢

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

⁢

π

=

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

π

:

G

→

Q

, a

lift

of

Q

to

G

is a function

f

:

Q

→

G

such that

π

⁢

f

=

1

Q

.

It follows that

π

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

π

:

G

→

Q

, a

splitting map

of

Q

to

G

is a

homomorphism

f

:

Q

→

G

such that

π

⁢

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

π

:

G

→

Q

,

f

:

Q

→

G

satisfying

π

⁢

f

=

1

Q

.

It follows,

f

is injective and

π

is

surjective

.

•

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

structure

involved.

Definition 5

.

Given a group

G

=

⟨

S

⟩

, there is a natural map

π

:

F

⁢

(

S

)

→

G

from the

free group

on

S

onto

G

. A

lift

is a map

l

:

G

→

F

⁢

(

S

)

such that

π

⁢

l

=

1

G

. Furthermore a

sift

is a lift

s

:

G

→

F

⁢

(

S

)

with the added condition that

s

⁢

g

=

g

for all

g

∈

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

⁢

⋯

⁢

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

⁢

⋯

⁢

t

n

-

1

for unique

t

i

∈

T

i

.

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

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