Group homomorphism

A function between groups that preserves the group operation: the image of a product equals the product of the images. It consequently maps the identity to the identity and inverses to inverses.

Group: A set equipped with an associative binary operation, an identity element, and an inverse for every element. Homomorphisms compare groups by preserving their operations.
First isomorphism theorem: A theorem stating that a group modulo a homomorphism's kernel is isomorphic to its image. It captures how the kernel and image determine the map's effective structure.
Group representation: A homomorphism from a group to a group of invertible linear transformations. Representations turn abstract group operations into transformations of vector spaces.
Group isomorphism: A bijective group homomorphism whose inverse is also a homomorphism. An isomorphism preserves operations and loses no information; a general homomorphism may collapse elements.
Binary operation: A rule that combines two elements of a set to produce another element of that set. The operation is the structure a group homomorphism must preserve.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. Every group homomorphism's kernel is a normal subgroup.
Permutation representation: A homomorphism from a group into a permutation group. It realizes each group element as a permutation while preserving multiplication.
Group monomorphism: An injective group homomorphism that identifies its domain with a subgroup of its codomain. Injectivity is an additional property, not part of the general homomorphism definition.
Identity element: An element that leaves every element unchanged when combined with it under a binary operation. A group homomorphism sends the source identity to the target identity.
Subgroup: A subset of a group that is itself a group under the inherited operation. A homomorphism maps subgroups to subgroups, including its image.

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