Uniqueness of Group Presentations

[Newtime]

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?

 

[fresh_42]

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.