The definition of a group $G$ includes the associativity of the operation. Then there is a requirement for the existence of an identity element. That is, there exists an element $e$ such that for all $a\in G$ $ae = ea = a$.
The next thing on the list in the definition is that for all $a\in G$ there exists an element $b\in G$ such that $ab = ba = e$.
Now after the definition we can prove that there is only one identity and that each element only has one inverse.
My question is how to read the definition if one doesn't know that there is only one identity element. Does the third property say that for each identity element $e\in G$ and for each $a\in G$ there is an element $b\in G$ such that $ab =ba = e$?
Or is this the requirement in the third property only that an element has an inverse for one identity element?
That is, is it
- For all $a\in G$ and for all identity elements $e\in G$ there is a $b\in G$ such that $ab =ba = e$
- For one fixed identity element $e\in G$ such that for all $a\in G$ there is a $b\in G$ such that $ab =ba = e$.
Again, I know that it doesn't matter since the two are equivalent. My question is what technically the definition says.
An identity is guaranteed to exist. You pick one, $e$, and the axiom of existence of inverse is with respect to $e$ (so it's the second one). Actually, formally, a group is a triplet $(G,e,\star)$, where $e$ is a given identity and $\star$ is the operation.