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Isoclinism
In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in and . The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope. Some textbooks discussing isoclinism include and and . Definition The isoclinism class of a group ''G'' is determined by the groups ''G''/''Z''(''G'') (the inner automorphism group) and ''G''′ (the commutator subgroup) and the commutator map from ''G''/''Z''(''G'') × ''G''/''Z''(''G'') to ''G''′ (taking ''a'', ''b'' to ''aba−1b−1''). In other words, two groups ''G''1 and ''G''2 are isoclinic if there are isomorphisms from ''G''1/''Z''(''G''1) to ''G''2/''Z''(''G''2) and from ''G''1′ to ''G''2′ commuting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
