Sylow subgroup of normal subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
===Symbol-free definition=== | |||
A [[subgroup]] of a [[finite group]] is termed a '''Sylow subgroup of normal subgroup''' if it satisfies the following equivalent conditions: | A [[subgroup]] of a [[finite group]] is termed a '''Sylow subgroup of normal subgroup''' if it satisfies the following equivalent conditions: | ||
# It is a [[Sylow subgroup]] of a [[normal subgroup]] of the whole group. | |||
# It is the intersection of a [[normal subgroup]] of the whole group with a [[Sylow subgroup]] of the whole group. | |||
# It is a Sylow subgroup inside its [[normal closure]]. | |||
===Equivalence of definitions=== | |||
{{further|[[Equivalence of definitions of Sylow subgroup of normal subgroup]]}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 13:13, 28 September 2008
This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: Sylow subgroup and normal subgroup
View other such compositions|View all subgroup properties
Definition
Symbol-free definition
A subgroup of a finite group is termed a Sylow subgroup of normal subgroup if it satisfies the following equivalent conditions:
- It is a Sylow subgroup of a normal subgroup of the whole group.
- It is the intersection of a normal subgroup of the whole group with a Sylow subgroup of the whole group.
- It is a Sylow subgroup inside its normal closure.
Equivalence of definitions
Further information: Equivalence of definitions of Sylow subgroup of normal subgroup