Let be a group and let .

- Prove that if is characteristic in and is normal in , then is normal in .
- Prove that if is characteristic in and is characteristic in , then is characteristic in . Use this to prove that the Klein 4-group is characteristic in .
- Give an example to show that if is normal in and is characteristic in then need not be normal in .
- Let denote the automorphism of which conjugates by . Since is normal in , we have ; thus is an automorphism of . Since is characteristic in , we thus have for all . Hence is normal in .
- Let be an automorphism of . Since is characteristic in , we have ; thus is an automorphism of . Since is characteristic in , we have ; thus is characteristic in .
Now is the unique subgroup of order 4 in , and is the unique subgroup of order 12 in . Thus and are characteristic; hence is characteristic in .

- Recall the subgroup lattice of . We have characteristic since is the unique subgroup of order 4, and is normal because is abelian. However, letting and , note that . Thus is not characteristic in .

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