how to prove homomorphism with identity element

Question

I have this,
Let $\phi: G → G$ defined by $\phi(x) = e$ , for all $x\in G$ ,where $e$ is the identity element. How to show that $\phi$ is a homomorphism?

Answer 1

We need to see that $\phi(xy) = \phi(x)\phi(y)$ for all $x,y \in G$. But $\phi(xy) = e = e\cdot e = \phi(x)\phi(y)$. So $\phi$ is a homomorphism.

These kinds of early group theory proofs about the identity pretty much always rely on the fact that $e\cdot e = e$. Another similar problem is to prove that the identity is unique -- i.e., if there are two elements $e$ and $f$ in $G$ with $ex = xe = x$ and $fx = xf = x$ for all $x \in G$, then $e=f$. See if you can work that one out too.