Characteristicity is quotient-transitive
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)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about quotient-transitive subgroup
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 .
- Normality is quotient-transitive
- Full invariance is quotient-transitive
- Strict characteristicity is quotient-transitive
Given: A group , subgroups such that is characteristic in , and is characteristic in
To prove: is characteristic in
Proof: We pick any automorphism of , and want to show that . For this, 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, . Since the same holds for , we conclude that , completing the proof.