Normal Sylow subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and Sylow subgroup
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Definition

A subgroup of a finite group is termed a normal Sylow subgroup if it satisfies the following equivalent conditions:

• It is a Sylow subgroup, and is normal in the whole group.
• It is a Sylow subgroup, and is subnormal in the whole group.
• It is a Sylow subgroup, and is characteristic in the whole group.
• It is a Sylow subgroup, and is fully characteristic in the whole group.

Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property Sylow subgroup
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

• Sylow direct factor

Weaker properties

• Nilpotent normal subgroup
• Nilpotent characteristic subgroup
• Normal Hall subgroup
• Complemented normal subgroup
• Fully characteristic subgroup
• Characteristic subgroup
• Intermediately characteristic subgroup
• Isomorph-free subgroup
• Intermediately fully characteristic subgroup
• Image-closed characteristic subgroup
• Image-closed fully characteristic subgroup
• Normal subgroup

Metaproperties

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
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Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition