Component (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 (Aschbacher & Seitz 1976).

References

  • Aschbacher, Michael (2000), Finite Group Theory, Cambridge University Press, ISBN 978-0-521-78675-1 
  • Aschbacher, Michael; Seitz, Gary M. (1976), "On groups with a standard component of known type", Osaka J. Math. 13 (3): 439–482 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Glossary of group theory — A group ( G , •) is a set G closed under a binary operation • satisfying the following 3 axioms:* Associativity : For all a , b and c in G , ( a • b ) • c = a • ( b • c ). * Identity element : There exists an e ∈ G such that for all a in G , e •… …   Wikipedia

  • Component — may refer to: Usage Component may refer to: System components, the constituents of a system Electronic components, the constituents of electronic circuits Component ingredient, the main ingredient in a dish Component video, a type of analog video …   Wikipedia

  • Theory of mind — is the ability to attribute mental states beliefs, intents, desires, pretending, knowledge, etc. to oneself and others and to understand that others have beliefs, desires and intentions that are different from one s own.[1] Though there are… …   Wikipedia

  • Group psychotherapy — is a form of psychotherapy in which one or more therapists treat a small group of clients together as a group. The term can legitimately refer to any form of psychotherapy when delivered in a group format, including Cognitive behavioural therapy… …   Wikipedia

  • Group delay and phase delay — Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component. Phase delay is a similar measure of the time… …   Wikipedia

  • Group method of data handling — (GMDH) is a family of inductive algorithms for computer based mathematical modeling of multi parametric datasets that features fully automatic structural and parametric optimization of models. GMDH is used in such fields as data mining, knowledge …   Wikipedia

  • Theory of Deep Democracy — Theory of Deep DemocracyThe theory of deep democracy makes a distinction between merely formal and deeper forms of democracy. Formal democracy is an important part of deep democracy, but it is merely a beginning or a necessary condition. In order …   Wikipedia

  • Lorentz group — Group theory Group theory …   Wikipedia

  • Orthogonal group — Group theory Group theory …   Wikipedia

  • General linear group — Group theory Group theory …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”