Quotient group

A quotient group is the group of cosets of a normal subgroup, with multiplication induced by the original group operation. It compresses a group by treating elements in the same coset as equivalent.

Coset: A subset formed by multiplying every element of a subgroup by a fixed group element. Each element of the quotient group is a coset of the subgroup.
Group: A set with an associative operation, an identity element, and an inverse for every element. A quotient group inherits its operation’s group laws from the original group.
Cyclic group: A group generated by powers of a single element. Quotients of cyclic groups remain cyclic, making them basic examples.
Left coset space: The set of left cosets of a subgroup in a group, without necessarily carrying a group structure. Cosets alone form a set; they form a quotient group only when the subgroup is normal.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. Normality ensures coset multiplication does not depend on the chosen representatives.
Subgroup: A subset of a group that is itself a group under the inherited operation. The subgroup being collapsed must first be a subgroup of the original group.
Integers modulo n: The additive group of integer residue classes modulo a positive integer n. It is the quotient of the integers by the subgroup of multiples of n.
Right coset: A subset formed by multiplying a subgroup on the right by a fixed group element. Left and right cosets need not agree unless the subgroup is normal.
Group homomorphism: A function between groups that preserves their group operation. A homomorphism’s kernel determines a quotient that captures its image.
Equivalence relation: A relation that is reflexive, symmetric, and transitive, partitioning a set into classes. Membership in the same coset defines the equivalence classes used in the quotient.

Linked from 23 pages