Quotient group

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Definition

Let $(G,\times)$ be a group. Let $H\subseteq G$ be a normal subgroup of $G$. Then:

• the cosets of $H$ in $G$ form a group whose operation is the group operation on subsets

We denote this new group $\frac{G}{H}$ or $G/H$.

With it we also get a group homomorphism called the canonical projection of the quotient group:

• $\pi:G\rightarrow\frac{G}{H}$ given by $\pi:g\rightarrow [g]$ where $[g]$ denotes the coset containing $g$.

Caution:This requires some work, why must the cosets of $H$ in $G$ partition $G$?

See also

• Normal subgroup
  • Every subgroup of an abelian group is normal