# Quotient group

*of a group by a normal subgroup *

The group formed by the cosets (cf. Coset in a group) , , of ; it is denoted by (cf. Normal subgroup). Multiplication of cosets is performed according to the formula

The unit of the quotient group is the coset , and the inverse of the coset is .

The mapping is a group epimorphism of onto , called the canonical epimorphism or natural epimorphism. If is an arbitrary epimorphism of onto a group , then the kernel of is a normal subgroup of , and the quotient group is isomorphic to ; more precisely, there is an isomorphism of onto such that the diagram

is commutative, where is the natural epimorphism .

A quotient group of a group can be defined, starting from some congruence on (cf. Congruence (in algebra)), as the set of classes of congruent elements relative to multiplication of classes. All possible congruences on a group are in one-to-one correspondence with its normal subgroups, and the quotient groups by the congruences are the same as those by the normal subgroups. A quotient group is a normal quotient object in the category of groups.

#### Comments

#### References

[a1] | P.M. Cohn, "Algebra" , I , Wiley (1982) pp. Sect. 9.1 |

**How to Cite This Entry:**

Quotient group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quotient_group&oldid=16178