Cyclic normal implies hereditarily normal
From Groupprops
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 normal subgroup) must also satisfy the second subgroup property (i.e., hereditarily normal subgroup)
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
View other applications of characteristic of normal implies normal OR Read a survey article on applying characteristic of normal implies normal
Contents
Statement
Verbal statement
Any subgroup of a cyclic normal subgroup is normal. (Also, any subgroup of a cyclic normal subgroup is cyclic, so in fact, subgroups of cyclic normal subgroups are cyclic normal).
Property-theoretic statement
The subgroup property of being a cyclic normal subgroup is stronger than the subgroup property of being a hereditarily normal subgroup. Equivalently, the property of being cyclic normal is a left-hereditary subgroup property.
Facts used
- Any subgroup of a cyclic group is a characteristic subgroup thereof
- A characteristic subgroup of a normal subgroup is normal
Proof
Given: A group , a cyclic normal subgroup , and a subgroup of
To prove: is normal in
Proof: By fact (1), and the given fact that is cyclic, is characteristic in . By fact (2), and the given datum that is normal in , we conclude that is normal in .
References
Textbook references
- Topics in Algebra by I. N. Herstein, ^{More info}, Page 53, Problem 13