Yes, it is sufficient to say that a group is commutative (or abelian) if there exists a group homomorphism that guarantees the condition $e * g = g * e$ for all $g \in G$.

Let's break this down:

1. Group Homomorphism: A group homomorphism $\varphi: G \to G$ is a map that respects the group operation. That is, for all elements $a, b \in G$, it satisfies: $\varphi(a * b) = \varphi(a) * \varphi(b)$ Now, you are given that there exists a homomorphism $\varphi$ such that $\varphi(g) = e$ for all $g \in G$, which implies that each element $g$ in $G$ behaves like the identity element under this homomorphism.

2. Condition $e * g = g * e$: The condition that $e * g = g * e$ for all $g \in G$ means that $g$ commutes with the identity element $e$. This is indeed true for all groups since the identity element satisfies $e * g = g * e = g$ for any $g$.

3. Commutative Groups: For a group to be commutative (abelian), the defining property is that the group operation satisfies: $g * h = h * g \quad \text{for all} \quad g, h \in G$ If the condition that $e * g = g * e$ holds universally for all $g \in G$, and it is established that the identity $e$ behaves in a way that helps show commutativity, you are essentially verifying that the group satisfies the commutative property.

Thus, your homomorphism gives a strong indication that $G$ might be commutative, provided it enforces the operation $e * g = g * e$ for all $g$. This hints that the group operation behaves similarly to an abelian group, but extra care should be taken to check if the entire group structure actually commutes (i.e., check whether $g * h = h * g$ for all $g, h$).

Follow-up:

Would you like to dive deeper into understanding group homomorphisms or the specific properties of abelian groups? Here's some additional exploration:

1. How do group homomorphisms relate to group isomorphisms?
2. What is the relationship between the identity element and commutativity in a group?
3. Could there be homomorphisms that don't imply commutativity?
4. How does the commutative property simplify group theory proofs?
5. Can a non-abelian group have any homomorphisms with trivial behavior like this?

Tip: The key to proving commutativity in some groups is often to look for commutator relations and analyze how the identity element and group elements interact with each other.