# Quotient group

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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.