Consider the following group in presentation form

In summary, the conversation discusses the presentation of a group with 2 generators that is commutative and shrinks any even power of one of the generators to the identity. The presentation can also be written in a different way, which may not resemble the group \mathbb{Z}/2\times \mathbb{Z}. The group is also discussed in terms of its formal relations and the concept of associating generators with their inverses. There is a clarification about the relation a=a^-1 and the problem with trying to apply it to the group \mathbb{Z}/2\times \mathbb{Z}.
  • #1

quasar987

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Something's bugging me. Consider the following group in presentation form:

[tex]<a,b|aba^{-1}b^{-1}=e,a^2=e>[/tex]

Ok, this is the presentation of a group with 2 generators that is commutative and "shrinks" any even power of one the generators to the identity. This sounds like an appropriate presentation of [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex].

But the presentation can also be written in another way:

[tex]<a,b|aba^{-1}b^{-1}=e,a^2=e>=<a,b|ab=ba,a=a^{-1}>[/tex]

Now this does not resembles [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex]! It says that [itex]<a,b|ab=ba,a=a^{-1}>[/itex] is a 2 generator abelian group that associates one of its generator with its inverse. Maybe I'm just tired but how does this apply to [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex]??
 
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  • #2
a=a^-1 is precisely the same as saying a^2=e. What's the problem?

the relations do not 'associate' things. They are formal relations that a and b (in this case) satisfy (and are the only such, except for those that can be deduced from them).
 
  • #3
I see it now. I was just tired. :approve:
 

What is a group in presentation form?

A group in presentation form is a way of representing a group using a list of generators and relations. It is a concise and efficient way to describe the elements and operations of a group.

How is a group presented?

A group is presented by listing its generators, which are the elements that can be combined to create all other elements in the group, and its relations, which are the equations that must hold true for the group's operations.

What are the benefits of using presentation form for a group?

Presentation form allows for a clear and concise representation of a group, making it easier to understand and work with. It also allows for efficient computation and analysis of the group's properties and elements.

Can any group be presented?

Not all groups can be presented in a finite way. Some groups may have an infinite number of generators and/or relations, making it impossible to list them all.

How is a group's order determined from its presentation?

The order of a group can be determined by counting the number of elements that can be generated from the given set of generators and relations. This can be done using mathematical techniques such as coset enumeration or the Todd-Coxeter algorithm.

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