# What is the factor group $\mathbb{Z}/5\mathbb Z$?

I am trying to understand the concept of factor group. The definition of factor group I know is the following: Let $G$ be a group and $H$ be a subgroup of $G$. Then the group of cosets denoted by $G/H$ is called the factor group of $G$ by $H$. Now I am looking at an example: Let $G=\mathbb{Z}$ and $H=5\mathbb{Z}$. Then I find the cosets of $5\mathbb{Z}$. Here they are:

• Cosets of $5\mathbb{Z}$ containing $0$, $\{\ldots,-10,-5,0,5,10,\ldots\}$
• Cosets of $5\mathbb{Z}$ containing $1$, $\{\ldots,-9,-4,1,6,11,\ldots\}$
• Cosets of $5\mathbb{Z}$ containing $2$, $\{\ldots,-8,-3,2,7,12,\ldots\}$
• Cosets of $5\mathbb{Z}$ containing $3$, $\{\ldots,-7,-2,3,8,13,\ldots\}$
• Cosets of $5\mathbb{Z}$ containing $4$, $\{\ldots,-6,-1,4,9,14,\ldots\}$

My question is, is the factor group $\mathbb{Z}/5\mathbb{Z}$ the union of all these $5$ cosets? Then I think $G/5\mathbb{Z} =\mathbb{Z}$ since the union of these cosets are equal to $\mathbb{Z}$. Can anyone help me with this?

Thanks.

The factor group is not the union of cosets, it's the set of cosets, so $${\bf Z}/5{\bf Z}=\{5{\bf Z},1+5{\bf Z},2+5{\bf Z},3+5{\bf Z},4+5{\bf Z}\}$$
The union of cosets is the original group. And $H$ has to be a normal subroup for the set of cosets to be a group (this is trivial if $G$ is abelian, of course).