What's the difference between identity and null element?

Question

I have the basic ideas about identity and null element for $\mathbb{R}$ numbers. I want to know if the same idea can be used for $\textbf{group theory}$ and what things I $\textbf{should consider}$.

Answer 1

If the group $(G,\circ)$ has addition as composition, i.e., for $x\circ y=x+y$, then $0$ is the neutral element. If $G$ has multiplication as composition, i.e., $x\circ y=xy$, then $1$ is the neutral element. For the field of real numbers we have in particular two groups: $G=(\mathbb{R},+)$ and $G'=(\mathbb{R}^*,\cdot)$. In the second, $1$ is the neutral element, the identity element. The requirement in general in $G$ for a neutral element is $$ x\circ e=e\circ x=x $$ for all $x\in G$.