What does $\mathbb{Z}/n \mathbb{Z}$ mean in abstract algebra? If you look at this wiki page under the image on top of right hand side, you see $\mathbb{Z}/ \mathbb{8Z}$. What does it mean and give example if possible please thanks.
 A: $\Bbb Z / 8\Bbb Z$ means the group of integers modulo $8$. (Up to isomorphism) it consists of the eight elements
$$
\{0, 1, 2, 3, 4, 5, 6, 7\}
$$
and the group operation is addition and then, if possible, subtraction by $8$. For instance, $1+3 = 4$, and $6+7 = 5$ ($13$ becomes $5$ after subtracting $8$).
This may be done for any integer $n$. We have $\Bbb Z/3\Bbb Z$ which consists of the elements $\{0, 1, 2\}$ with the addition table
$$
\begin{array}{c|ccc}
+&0&1&2\\
\hline0&0&1&2\\
1&1&2&0\\
2&2&0&1
\end{array}
$$
and $\Bbb Z/186\,432\,515\,384\,315\Bbb Z$ which is quite large, but just as straight-forward.
A: Since you mentioned Lagrange's theorem, I assume we are talking about groups.
$\mathbb{Z}$ is the group of integers with addition as group operation.
$8\mathbb{Z}$ is the set $\{8k | k \in \mathbb{Z}\}$, which is the set of all integer multiples of $8$. It is a subgroup, and a normal one (since the group is abelian, any subgroup is normal).
Now, $\mathbb{Z}/8\mathbb{Z}$ means the quotient group of $\mathbb{Z}$ modulo the subgroup $8\mathbb{Z}$, and is a cyclic group with $8$ elements.
