Subgroup properties | Group theory

Component (group theory)

In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of the group. For finite abelian (or nilpotent) groups, p-component is used in a different sense to mean the Sylow p-subgroup, so the abelian group is the product of its p-components for primes p. These are not components in the sense above, as abelian groups are not quasisimple. A quasisimple subgroup of a finite group is called a standard component if its centralizer has even order, it is normal in the centralizer of every involution centralizing it, and it commutes with none of its conjugates. This concept is used in the classification of finite simple groups, for instance, by showing that under mild restrictions on the standard component one of the following always holds: * a standard component is normal (so a component as above), * the whole group has a nontrivial solvable normal subgroup, * the subgroup generated by the conjugates of the standard component is on a short list, * or the standard component is a previously unknown quasisimple group. (Wikipedia).

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From playlist Abstract algebra

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From playlist Essence of Group Theory

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From playlist Modern Algebra - Chapter 15 (groups)

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From playlist Group theory

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From playlist Group theory

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From playlist Global Noncommutative Geometry Seminar (Americas)

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From playlist Seminar on Geometric and Modular Representation Theory

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Quotient group example

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From playlist Abstract algebra

Related pages

Involution (mathematics) | Sylow theorems | Normal subgroup | Classification of finite simple groups | Mathematics | Solvable group | Fitting subgroup | Nilpotent group | Quasisimple group | Group theory | Conjugacy class | Subnormal subgroup | Abelian group | Finite group | Group (mathematics)