Fundamental theorem on homomorphisms

For a group homomorphism, the quotient of its domain by its kernel is isomorphic to its image. It identifies elements that map to the same value.

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Group homomorphism: A function between groups that preserves the group operation. The theorem describes the structure preserved by this kind of map.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. A homomorphism's kernel is normal, so its cosets form a quotient group.
Group action: A way for a group to act as symmetries on a set while respecting group multiplication. Orbit maps often yield quotient descriptions through their kernels.
Second isomorphism theorem for groups: A theorem relating a subgroup and a normal subgroup through their intersection and product. It is a neighboring isomorphism theorem, but compares subgroups rather than a map's image.
Kernel of a group homomorphism: The subgroup of a group homomorphism's domain whose elements map to the identity. Its cosets are precisely the classes identified by the theorem.
Coset: A translate of a subgroup within a group, formed by multiplying its elements by one group element. Each kernel coset gathers domain elements with the same image.
Orbit-stabilizer theorem: A theorem equating an orbit's size with the index of the point's stabilizer in a group. The orbit map is a homomorphism from a group quotient onto an orbit when the action is transitive.
Third isomorphism theorem for groups: A theorem identifying a quotient of a quotient group with a single quotient by a larger normal subgroup. It simplifies nested quotients instead of describing a homomorphism's image.
Quotient group: A group formed from the cosets of a normal subgroup, with an induced group operation. Quotienting by the kernel creates the theorem's domain-side group.
Group isomorphism: A bijective group homomorphism whose inverse also preserves the group operation. The theorem concludes that the quotient and image have identical group structure.