Quotient Group

For a Group $G$ and a Normal Subgroup $N$ of $G$, the quotient group of $N$ in $G$, written $G/N$ and read ``$G$ modulo $N$'', is the set of Cosets of $N$ in $G$. The elements of $G/N$ are written $Na$ and form a Group under the normal operation on the group $N$ on the coefficient $a$. Thus,

$$(Na)(Nb)=Nab.$$

Since all elements of $G$ will appear in exactly one Coset of the Normal Subgroup $N$, it follows that

$$\vert G/N\vert=\vert G\vert/\vert N\vert,$$

where $\vert G\vert$ is the order of the group $G$.

The slash Notation conflicts with that for a Field Extension, but the meaning can be determined based on context.

See also Abhyankar's Conjecture, Coset, Field Extension, Outer Automorphism Group, Normal Subgroup, Subgroup

References

Herstein, I. N. Topics in Algebra, 2nd ed. New York: Springer-Verlag, 1975.