• Source: Absolute presentation of a group

In mathematics, an absolute presentation is one method of defining a group.
Recall that to define a group



G


{\displaystyle G}

by means of a presentation, one specifies a set



S


{\displaystyle S}

of generators so that every element of the group can be written as a product of some of these generators, and a set



R


{\displaystyle R}

of relations among those generators. In symbols:




G
≃
⟨
S
∣
R
⟩
.


{\displaystyle G\simeq \langle S\mid R\rangle .}


Informally



G


{\displaystyle G}

is the group generated by the set



S


{\displaystyle S}

such that



r
=
1


{\displaystyle r=1}

for all



r
∈
R


{\displaystyle r\in R}

. But here there is a tacit assumption that



G


{\displaystyle G}

is the "freest" such group as clearly the relations are satisfied in any homomorphic image of



G


{\displaystyle G}

. One way of being able to eliminate this tacit assumption is by specifying that certain words in



S


{\displaystyle S}

should not be equal to



1.


{\displaystyle 1.}

That is we specify a set



I


{\displaystyle I}

, called the set of irrelations, such that



i
≠
1


{\displaystyle i\neq 1}

for all



i
∈
I
.


{\displaystyle i\in I.}





Formal definition


To define an absolute presentation of a group



G


{\displaystyle G}

one specifies a set



S


{\displaystyle S}

of generators and sets



R


{\displaystyle R}

and



I


{\displaystyle I}

of relations and irrelations among those
generators. We then say



G


{\displaystyle G}

has absolute presentation




⟨
S
∣
R
,
I
⟩
.


{\displaystyle \langle S\mid R,I\rangle .}


provided that:




G


{\displaystyle G}

has presentation



⟨
S
∣
R
⟩
.


{\displaystyle \langle S\mid R\rangle .}


Given any homomorphism



h
:
G
→
H


{\displaystyle h:G\rightarrow H}

such that the irrelations



I


{\displaystyle I}

are satisfied in



h
(
G
)


{\displaystyle h(G)}

,



G


{\displaystyle G}

is isomorphic to



h
(
G
)


{\displaystyle h(G)}

.
A more algebraic, but equivalent, way of stating condition 2 is:

2a. If



N
◃
G


{\displaystyle N\triangleleft G}

is a non-trivial normal subgroup of



G


{\displaystyle G}

then



I
∩
N
≠

{
1
}

.


{\displaystyle I\cap N\neq \left\{1\right\}.}


Remark: The concept of an absolute presentation has been fruitful in fields such as algebraically closed groups and the Grigorchuk topology.
In the literature, in a context where absolute presentations are being discussed, a presentation (in the usual sense of the word) is sometimes referred to as a relative presentation, which is an instance of a retronym.




Example


The cyclic group of order 8 has the presentation




⟨
a
∣

a

8


=
1
⟩
.


{\displaystyle \langle a\mid a^{8}=1\rangle .}


But, up to isomorphism there are three more groups that "satisfy" the relation




a

8


=
1
,


{\displaystyle a^{8}=1,}

namely:




⟨
a
∣

a

4


=
1
⟩


{\displaystyle \langle a\mid a^{4}=1\rangle }





⟨
a
∣

a

2


=
1
⟩


{\displaystyle \langle a\mid a^{2}=1\rangle }

and




⟨
a
∣
a
=
1
⟩
.


{\displaystyle \langle a\mid a=1\rangle .}


However, none of these satisfy the irrelation




a

4


≠
1


{\displaystyle a^{4}\neq 1}

. So an absolute presentation for the cyclic group of order 8 is:




⟨
a
∣

a

8


=
1
,

a

4


≠
1
⟩
.


{\displaystyle \langle a\mid a^{8}=1,a^{4}\neq 1\rangle .}


It is part of the definition of an absolute presentation that the irrelations are not satisfied in any proper homomorphic image of the group. Therefore:




⟨
a
∣

a

8


=
1
,

a

2


≠
1
⟩


{\displaystyle \langle a\mid a^{8}=1,a^{2}\neq 1\rangle }


Is not an absolute presentation for the cyclic group of order 8 because the irrelation




a

2


≠
1


{\displaystyle a^{2}\neq 1}

is satisfied in the cyclic group of order 4.




Background


The notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem for algebraically closed groups.
A common strategy for considering whether two groups



G


{\displaystyle G}

and



H


{\displaystyle H}

are isomorphic is to consider whether a presentation for one might be transformed into a presentation for the other. However algebraically closed groups are neither finitely generated nor recursively presented and so it is impossible to compare their presentations. Neumann considered the following alternative strategy:
Suppose we know that a group



G


{\displaystyle G}

with finite presentation



G
=
⟨

x

1


,

x

2


∣
R
⟩


{\displaystyle G=\langle x_{1},x_{2}\mid R\rangle }

can be embedded in the algebraically closed group




G

∗




{\displaystyle G^{*}}

then given another algebraically closed group




H

∗




{\displaystyle H^{*}}

, we can ask "Can



G


{\displaystyle G}

be embedded in




H

∗




{\displaystyle H^{*}}

?"
It soon becomes apparent that a presentation for a group does not contain enough information to make this decision for while there may be a homomorphism



h
:
G
→

H

∗




{\displaystyle h:G\rightarrow H^{*}}

, this homomorphism need not be an embedding. What is needed is a specification for




G

∗




{\displaystyle G^{*}}

that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.




References

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