Cyclic characteristic implies hereditarily characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cyclic characteristic subgroup) must also satisfy the second subgroup property (i.e., hereditarily characteristic subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about cyclic characteristic subgroup|Get more facts about hereditarily characteristic subgroup
Suppose is a group and is a cyclic characteristic subgroup of , i.e., is a cyclic group and is also a Characteristic subgroup (?) of . Then, is also a hereditarily characteristic subgroup of , i.e., every subgroup of is characteristic in .
- Cyclic normal implies hereditarily normal
- Hereditarily characteristic not implies cyclic in finite
- Cyclic-quotient characteristic implies upward-closed characteristic
- Cyclic implies every subgroup is characteristic
- Characteristicity is transitive: If with characteristic in and characteristic in , then is characteristic in .
Given: A group with a cyclic characteristic subgroup . A subgroup of .
To prove: is characteristic in .
- is characteristic in : This follows from fact (1).
- is characteristic in : This follows from the previous step, the given datum that is characteristic in , and fact (2).