Definition of Existence of Identity in a Group

Question

For a binary operation structure to be a group in a Set, i.e., $G = \langle*,S\rangle$

One must have an identity $e$ in $G$ s.t. $a*e = e*a =a \ \forall a \in G$.

But does it mean that e is one element which is applied to all element in $G$ or just it's okay for $e$ to specifically exist for each $a$ in $G$, e.g., if $(e_0, a_0), (e_1, a_1)\cdots (e_n, a_n)$ where $e_i$'s are different and meeting the criterion $e_i * a_i = a_i * e_i = a_i$ ?

Answer 1

It's important to write this out in quantifiers:

$$\exists e \in G: \forall g \in G: ge =g = eg$$

In other words, the existential quantifier comes for the universal quantifier, and hence it means that the element $e$ has to work for every group element.

In particular, it follows that this element $e$ is unique so the answer to your question is no: we can't have different identities for different elements. Every element has the same identity.

Answer 2

$e$ is supposed to be one element that serves as the identity for the entire group.

Answer 3

The identity should work for all elements and it is unique.

That means if $e_1$ and $e_2$ are identities then $e_1=e_2$

Just like for the group of integers under addition, there is only one $0$ which works for all integers.

Answer 4

It is just one $e$ that satisfies this criterion. Identity doesn't have to specifically exist for each and every element. Try proving the theorem:the identity element in a group is unique Hint: assume it there are two identities and go on to show that they both are the same.