The Group Properties Wiki (pre-alpha)

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From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and Sylow subgroup
View other subgroup property conjunctions | view all subgroup properties

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Transfer condition 3.2 Intermediate subgroup condition 3.3 Image condition

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.

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

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.
View a complete list of such properties

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition

Image condition

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 a complete list of subgroup properties satisfying the image condition