Contracharacteristic subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed a contracharacteristic subgroup if it satisfies the following equivalent conditions:

1. Its characteristic closure in the whole group equals the whole group.
2. It is not contained in any proper characteristic subgroup of the whole group.

Relation with other properties

Stronger properties

• Contranormal subgroup

Facts

• Every subgroup is contracharacteristic in its normal closure: In particular, every subgroup of a group is a contracharacteristic subgroup of a normal subgroup.