Group theory MOC

Quotient group

Given a normal subgroup , the quotient group is a group of the cosets of , defined as follows group

with the canonical projection

The quotient group is the natural application of the Algebraic quotient in the group context. However, instead of taking the quotient mod a congruence relation, it is typical to use the corresponding normal subgroup. Hence may alternatively be referred to as , taken to mean the equivalence class of under the congruence induced by . Another notation is to just use the elements of but replace with .

Universal property

The quotient group with the canonical projection is characterized up to unique isomorphism by the universal property:

. If is a group and is a homomorphism with , then there exists a unique homomorphism so that , i.e.

Proof

By construction, . A homomorphism can be factored via iff , and this holds iff . The uniqueness of follows from being an epimorphism: . Therefore fulfils the universal property. If also fulfils the universal property, then the following diagram commutes:

giving the required unique isomorphism.

Properties

• By Lagrange’s theorem .

Special quotients

• Abelianization

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