Regarding group notation

Question

I came across the following notation and couldn't make sense of it: $\mathbb{Z}/p\mathbb{Z}$

What is this denoting? What are the elements of this set?

Answer 1

This is the collection of sets $$\{n+p\mathbb Z\mid n\in\mathbb Z\}$$ which forms a group $(G,+_G)$ with the group operation “$+_G$” defined by $$(n+p\mathbb Z)+_G (m+p\mathbb Z)=(n+m)+p\mathbb Z$$

Recall that $$p\mathbb Z=\{\ldots, -2p,-p,0,p,2p,\ldots\}$$ and that

$$n+p\mathbb Z=\{\ldots, n-2p,n-p,n,n+p,n+2p,\ldots\}$$

For a given positive integer $p$, this group consists of exactly $p$ distinct elements. For example, consider $p=3$. Then $$0+3\mathbb Z=\{\ldots,-6,-3,0,3,6,\ldots\}$$

$$1+3\mathbb Z=\{\ldots,-5,-2,1,4,7,\ldots\}$$

$$2+3\mathbb Z=\{\ldots,-4,-1,2,5,8,\ldots\}$$ After this, they repeat: $$3+3\mathbb Z=\{\ldots,-3,0,3,6,9,\ldots\}=0+3\mathbb Z$$

$$4+3\mathbb Z=\{\ldots,-2,1,4,7,10,\ldots\}=1+3\mathbb Z$$

etc., so this group contains three elements: $$\mathbb Z/3\mathbb Z=\{3 \mathbb Z,1+ 3\mathbb Z,2+ 3\mathbb Z\}.$$

A standard exercise is to show that the group operation is well-defined in the sense that the particular representations of the elements (e.g., $1+ 3\mathbb Z$ vs $4 + 3\mathbb Z$) still gives the same result when performing the group addition.