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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.

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# Weak/strong group identity axiom

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