Uniqueness of Group Presentations

  • Thread starter Newtime
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Main Question or Discussion Point

I had a few most questions which should be trivial for the group theorists out there, but since I'm still relatively new to this, they have me stumped:

1. Given a presentation, how can one verify it is unique?
2. Given a presentation, how can one verify it is minimal aside from the obvious of manipulating relations into other relations?
3. Given a presentation, how can one verify that one has included all relations? In other words, how can one verify that a presentation is indeed a presentation?
 

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A group representation of a group ##G## is a group homomorphism ##\varphi\, : \,G \longrightarrow \operatorname{Aut}(H)## into an automorphism group of some set ##H##, which can be another group, or a vector space, in which we call the representation linear and the automorphism group ##\operatorname{GL}(V)##. This a group representation is the triple ##(G,\varphi,H)##.

It makes no sense to speak of uniqueness, since ##\varphi## is what makes it unique. Different homomorphisms mean different operations mean different representations.

The same goes for minimality. We can always define ##g.h :=h## i.e. map every element on the identity automorphism. This is automatically minimal in which sense you ever measure size.

Since ##\varphi## is required to be a group homomorphism, this is equivalent to the invariance of relations. One has to prove that ##\varphi## is a group homomorphism, so depending on how a representation is defined, the properties have to be proven, i.e. calculated.
 

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