Characteristicity is strongly intersection-closed

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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

Other particular cases of this general result are:

Analogues in other algebraic structures

Related metaproperty satisfactions and dissatisfactions for characteristicity

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

Proof

Hands-on proof

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Property-theoretic proof

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