Weak/strong group identity axiom

[matheinste]

Hello all.

While going back to group theory basics to make sure i understand rather than just know the fundamentals i came across for the first time ( having read many books ) the weak versus strong versions of the identity axiom. The strong version says that a group must have a unique identity element. The weak version says that there must be AN identity element for all members of the group. The uniqueness of the strong axiom identity can be proved from the weak version the associative and closure axioms. i understand this. So we only need to require the identity to satisfy the weak axiom as this implies the strong axiom. Fair enough. Then the author ( Alan F Beardon: Algebra and Geometry ) goes on to say that the weak version would make the final axiom, the inverse axiom, ambiguous. I can see that this is so if the weak version of the identity axiom were taken alone. But why is this so, as the author seems to imply, if the weak identity axiom implies the strong axiom.

Seems very basic but i would like to understand.

Thanks. Matheinste.

 

[morphism]

Why don't you post how the inverse axiom is stated in your book? Also, try to think what identity element they're using in that axiom.

 

[matheinste]

Hello Morphism.

In answer to your query the definition used is the standard ( only? ) one. I have yet to learn latex so will have to use words.

If g is in G there is an h in G such that gh=e=hg.

Matheinste

 

[morphism]

OK, and if e isn't unique, do you see how "gh=e=hg" is ambiguous?

 

[matheinste]

Thanks Morphism.

I do see why the weak identity axiom if it did not when used with the associative axiom and the axiom of closure lead to the strong identity axiom and thus the uniqueness of the inverse would lead to possible ambiguity in the inverse axiom ( we may have more than one e and hence more than one inverse ). But the fact that it ( the weak axiom ) does so surely removes any possible ambiguity.

It really does seem trivial i suppose. That is why i wondered why the authors ( more than one ) mention it. Or, in the often used phrase " am i missing something ".

Matheinste

Matheinste.