Sylow subgroup: Difference between revisions
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In [[group theory]], a '''Sylow subgroup''' of a [[group (mathematics)|group]] is a [[subgroup]] which has [[Order of a subgroup|order]] which is a power of a [[prime number]], and which is not strictly contained in any other subgroup with the same property. Such a subgroup may also be called a '''Sylow''' '''''p''''' '''-subgroup''' or a '''''p''''' '''-Sylow subgroup'''. | In [[group theory]], a '''Sylow subgroup''' of a [[group (mathematics)|group]] is a [[subgroup]] which has [[Order of a subgroup|order]] which is a power of a [[prime number]], and which is not strictly contained in any other subgroup with the same property. Such a subgroup may also be called a '''Sylow''' '''''p''''' '''-subgroup''' or a '''''p''''' '''-Sylow subgroup'''. | ||
Revision as of 12:55, 7 February 2009
In group theory, a Sylow subgroup of a group is a subgroup which has order which is a power of a prime number, and which is not strictly contained in any other subgroup with the same property. Such a subgroup may also be called a Sylow p -subgroup or a p -Sylow subgroup.
The Sylow theorems describe the structure of the Sylow subgroups. Suppose that p is a prime which divides the order n of a finite group G, so that , with t coprime to p
- Theorem 1. There exists at least one subgroup of G of order , which is thus a Sylow p-subgroup.
- Theorem 2. The Sylow p-subgroups are conjugate.
- Theorem 3. The number of Sylow p-subgroups is congruent to 1 modulo p.
The first Theorem may be regarded as a partial converse to Lagrange's Theorem.
References
- M. Aschbacher (2000). Finite Group Theory, 2nd ed, 19. ISBN 0-521-78675-4.