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Absolute presentation of a group

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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:

  1. G {\displaystyle G} has presentation ⟨ S ∣ R ⟩ . {\displaystyle \langle S\mid R\rangle .}
  2. 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.

See also

References

  1. ^ B. Neumann, The isomorphism problem for algebraically closed groups, in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562.
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