Meaning of $\setminus$ notation in Group Theory

Question

Usually $\setminus$ means "set minus", for example $A\setminus B$ means the elements of $A$ not in $B$.

But in one of my books on Complex Analysis, we have to show that the automorphisms of the unit disk, $\operatorname{Aut}(\mathbb{D})$, form a group. And that given the subgroup $K = \{ f \in \operatorname{Aut}(\mathbb{D}) : f(0)=0 \}$, we have $$ K \setminus \operatorname{Aut}(\mathbb{D}) \cong \mathbb{D} $$

My question is what does $ K \setminus \operatorname{Aut}(\mathbb{D}) $ stand for? It is not setminus, and it is not the quotient group (since the author uses quotient group in another context just before)

I have never seen this notation nor can I find it on the internet, please help!!

Answer 1

When the subgroup $H$ is not a normal subgroup of the group $G$, you can nevertheless define

• the left cosets of $H$ in $G=\{gH\mid g\in G\}$, which is a partition of $G$, and therefore associated to an equivalence relation on $G$. This set is denoted $G/H$;
• the right cosets of $H$ in $G=\{Hg\mid g\in G\}$, which is a partition of $G$, and associated to another equivalence relation on $G$. This set is denoted $H \backslash G$.

There are also the double cosets of two subgroups $H$ and $K$: $$ H\backslash G/K=\{HgK\mid g\in G\}.$$

Answer 2

Normally, if $H$ is a normal subgroup of $G$, then $$G/H=\{gH\mid g\in G\}\quad \text{and}\quad H\setminus G=\{Hg\mid g\in G\}.$$