Characteristicity is strongly intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)
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Statement

Verbal statement

An arbitrary (possibly empty) intersection of characteristic subgroups of a group is a characteristic subgroup.

Statement with symbols

Suppose $I$ is an indexing set and $H_i, i \in I$ is a collection of characteristic subgroups of a group $G$. Then, the intersection of subgroups $\bigcap_{i \in I} H_i$ is also a characteristic subgroup of $G$.

Related facts

Generalizations

• Invariance implies strongly intersection-closed: Any invariance property (i.e., a property that can be expressed as invariance under a certain collection of functions) is strongly intersection-closed: an arbitrary intersection of subgroups with the property again has the property.

Other particular cases of this general result are:

• Normality is strongly intersection-closed
• Strict characteristicity is strongly intersection-closed
• Full characteristicity is strongly intersection-closed

Analogues in other algebraic structures

• Characteristicity is strongly intersection-closed in Lie rings
• Derivation-invariance is strongly intersection-closed

Related metaproperty satisfactions and dissatisfactions for characteristicity

• Characteristicity is transitive: A characteristic subgroup of a characteristic subgroup is characteristic.
• Characteristicity is strongly join-closed: The subgroup generated by a collection of characteristic subgroups is characteristic.
• Characteristicity is not finite-relative-intersection-closed
• Characteristicity does not satisfy intermediate subgroup condition
• Characteristicity does not satisfy transfer condition

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

Proof

Hands-on proof

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Property-theoretic proof

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