Q:

Closure, identity, and inverses in $\Bbb{Z}/6\Bbb{Z}$ as a group.

The group of integers mod $6$, $\Bbb{Z}/6\Bbb{Z}:={0,1,2,3,4,5}$ is a group under the action addition (+). Each of the $6$ elements of this group are associated with a unique action, $+0, +1, +2, +3, +4, +5$.
Every integer is associated with a single number in this set $\Bbb{Z}/6\Bbb{Z}$ which is the remainder after dividing by $6$. For example, the number $27$ is associated with $3$ since $27=4(6)+3$. We say "$27$ is equivalent to $3$ mod $6$".

Confirm that the set $\Bbb{Z}/6\Bbb{Z}:=\{0,1,2,3,4,5\}$ is closed under addition by listing all combinations of pairs of elements and showing that each result is equivalent to one of $0,1,2,3,4,5$.

What is the identity of this group?

Find the inverse of each element.

Thank you so much, I really appreciate any help :D
Stay safe guys!

From the comments . . .

I just dont know how to find the inverse or identity. I know that an identity is something that leaves other elements unchanged when combined with it, and I know that an inverse is like the "opposite", but I just don't know how to find them

A:

This can be done by way of computing the Cayley table, like so:
$$\begin{array}{c|cccccc}
+_6 & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
0 & 0 & 1 & 2 & 3 & 4 & 5 \\
1& 1&2&3&4&5&0\\
2&2&3&4&5&0&1\\
3&3&4&5&0&1&2\\
4&4&5&0&1&2&3\\
5&5&0&1&2&3&4
\end{array}.$$
Since each addition is equivalent to a number within $\{0,1,2,3,4,5\}$, the operation is closed. One can also see from the table that $0$ is the identity and that, here,
$$\begin{align}
-0&=0, \\
-1&=5, \\
-2&=4, \\
-3&=3, \\
-4&=2, \\
-5&=1.
\end{align}$$

