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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Conjunction with other properties 2.3 Incomparable properties

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Definition

Symbol-free definition

A subgroup of a group is termed core-characteristic if its normal core is a characteristic subgroup of the whole group.

Definition with symbols

A subgroup H of a group G is termed core-characteristic if the normal core H_G of H in G is a characteristic subgroup of G.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Automorph-conjugate subgroup
• Intersection of automorph-conjugate subgroups
• Core-free subgroup

Conjunction with other properties

Any normal subgroup that is also core-characteristic, is characteristic.

Incomparable properties

• Closure-characteristic subgroup