# Terminology of "G over H"

I am trying to find the definition of G/H (which is read as "G over H", "G modulo H", or "G mod H"). I believe that, in this case, G is a group and H is a subgroup of G.

• Do you know anything about quotient groups? This is what you're looking for. Oct 19, 2014 at 23:51
• – lhf
Oct 19, 2014 at 23:53

It depends if you are considering the quotient set, or the quotient group. Define: $$x \sim_L y \iff xy^{-1} \in H$$
We can also write this as $x \equiv_L y \mod H$. Here, $L$ is for "left". You can define it the same way for right. Then, we have the class: $$\bar{x}_L = \{y \in G \mid x \equiv_L y \mod H\}$$
Then, we define $G/H = \{ \bar{x}_L \mid x \in G\}$. If $H$ is normal in $G$, then in the end won't matter if we use the relation by the left or by the right, and we can define a operation in the quotient set, by: $$\bar{x} \cdot \bar{y} := \overline{xy}$$
It has to be proven that the operation does not depend on the representant of the class that you pick. Then $(G/H, \cdot)$ is called the quotient group.
Let $H$ be a normal subgroup of $G$. Then, the cosets of $H$ form a group $G/H$ under the binary operation $(aH)(bH)=(ab)H$.
The group $G/H$ is the factor group (or quotient group) of $G$ by $H$.