Question about Permutation group

Question

I have a doubt,

What is the identity element(for the group operator) for the permutation group defined over $N$ objects?

Also, What is the most elementary reason for the fact that the set of all possible permutations over $N$ objects along with the permutation operator is not a ring?

Answer 1

It is the identity map

$$\begin{align} \iota: \{1, \dots, N\} &\to\{1,\dots, N\},\\ n&\mapsto n. \end{align}$$

The reason it is not a ring is that there is no addition. What would be the additive identity?

Answer 2

Τhe identity element for the permutation group defined over $N$ objects $\{a_1,...a_N\}$ is the permutation $g$ defined by $g(a_i)=a_i \ \forall i\in\{1,...,N\}$.

In order to make the permutation group (symmetric group) into a ring you have to define addition over elements $g,g'$. Do you have any idea of how this could be achieved?