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P-nilpotent Group
In mathematical group theory, a normal p-complement of a finite group for a prime ''p'' is a normal subgroup of order coprime to ''p'' and index a power of ''p''. In other words the group is a semidirect product of the normal ''p''-complement and any Sylow ''p''-subgroup. A group is called p-nilpotent if it has a normal ''p''-complement. Cayley normal 2-complement theorem Cayley showed that if the Sylow 2-subgroup of a group ''G'' is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic. Burnside normal p-complement theorem showed that if a Sylow ''p''-subgroup of a group ''G'' is in the center of its normalizer then ''G'' has a normal ''p''-complement. This implies that if ''p'' is the smallest prime dividing the order of a group ''G'' and the Sylow ''p''-subgroup is cyclic, then ''G'' has a normal ''p''-complement. Frobenius normal p-complement theorem The Frobenius normal ''p''-complement the ... [...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] |
