So I'm very new to group theory and I'm trying to understand the difference between a coset and a quotiënt group. To me it seems that a quotient group is always also a coset but not the other way around. But I can't seem to grap the exact difference betweem them.
Can anyone give a simple example or intuitive explanation?
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$\begingroup$So you start with a group $G$ and a subgroup $H$.
Then the (left) coset of $H$ for $g\in G$ is a subset of $G$ of the form $\{gh\ |\ h\in H\}$. Also denoted by $gH$. Let me emphasize this one more time: cosets are certain special subsets of $G$.
The collection of all cosets is denoted by $G/H=\{gH\ |\ g\in G\}$. So elements of $G/H$ are cosets, not elements of $G$.
If $H$ is additionally a normal subgroup then $G/H$ can be naturally turned into a group as well and in this situation we call $G/H$ the quotient group.
Example:
Consider $G=\mathbb{Z}$ integers with addition and $H=5\mathbb{Z}$ i.e. all integers divisible by $5$. So every element of $H$ can be written as $5k$ for some $k\in\mathbb{Z}$. Therefore every coset can be written as $$x+H=\{x+5k\ |\ k\in\mathbb{Z}\}$$
So how many cosets are there? Since every coset is $x+H$ for some integer $x\in\mathbb{Z}$ then you may think that as many as there are integers, right? Wrong. For example
$$\{1+5k\ |\ k\in\mathbb{Z}\}=\{6+5t\ |\ t\in\mathbb{Z}\}$$
These are equal sets and so these two cosets: $1+H$ and $6+H$ are equal. You can verify that there are only 5 cosets of $H$, namely $H$, $1+H$, $2+H$, $3+H$ and $4+H$. They correspond to remainders to division by $5$. So $G/H$ has exactly $5$ elements even though $G$ is infinite.
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