Characteristicity is quotient-transitive

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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., 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 H \le K \le G are subgroups such that H is a characteristic subgroup of G, and K/H is a characteristic subgroup of G/H. Then, K is a characteristic subgroup of G.

Related facts

Similar facts

Proof

Given: A group G, subgroups H \le K \le G such that H is characteristic in G, and K/H is characteristic in G/H

To prove: K is characteristic in G

Proof: We pick any automorphism \sigma of G, and want to show that \sigma(K) = K. For this, first observe that \sigma(H) = H, so \sigma induces an automorphism on the quotient G/H, by the rule gH \mapsto \sigma(g)H. Call this automorphism \sigma'.

Then, \sigma' is an automorphism of G/H. Since K/H is characteristic in G/H, \sigma'(K/H) = K/H. Thus, for any g \in K, \sigma'(gH) \in K/H, and hence, unwrapping the definition, \sigma(g) \in K. Thus, \sigma(K) \subset K. Since the same holds for \sigma^{-1}, we conclude that \sigma(K) = K, completing the proof.