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Group homomorphismGiven two groups G ) and H ·), a group homomorphism from G ) to H ·) is a function h : G H such that for all u and v in G it holds that : h ''u v h ''u · h ''v From this property, one can deduce that h maps the identity element e of G to the identity elem Group isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, the GroupoidIn mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. They are often used to ca Groups and clusters of galaxiesGalaxy groups and clusters are super-structures in the spread of galaxies of the cosmos. Matter throughout the visible Universe has, over the course of the Universe's history, aggregated into a range of large-scale structures under the influence of gravit Group representationRepresentation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. Representation theory is important because it enables many group-theoretic problems to be r Group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitude (known as the envelope of the wave) propagates through space. The group velocity is defined in terms of the wave's angular frequency omega and wave number k Group action Group (mathematics) Groupuscule Group alerting and dispatching system Group delay Group theory Group process consultation Group Group therapy