Covering Groups Of The Alternating And Symmetric Groups
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for , the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold. For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4. Definition and classification A group homomorphism from ''D'' to ''G'' is said to be a Schur cover of the finite group ''G'' if: # the kernel is contained both in the center and the commutator subgroup of ''D'', and # amongst all such homomorphisms, this ''D'' has maximal size. The Schur multiplier of ''G'' is the kernel of any Schur cover and has many interpretations. When the homomorphism is under ...
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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 cen ...
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