Characteristicity is quotient-transitive

From Groupprops

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

Property-theoretic statement

The subgroup property of being a characteristic subgroup satisfies the subgroup metaproperty of being quotient-transitive.

Statement with symbols

Suppose are subgroups such that is a characteristic subgroup of , and is a characteristic subgroup of . Then, is a characteristic subgroup of .

Related facts

Similar facts

Proof

Given: A group , subgroups such that is characteristic in , and is characteristic in . An automorphism of .

To prove:

Proof: First observe that , so induces an automorphism on the quotient , by the rule . Call this automorphism .

Then, is an automorphism of . Since is characteristic in , . Thus, for any , , and hence, unwrapping the definition, . Thus, , completing the proof.