Abnormal implies WNSCC
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., abnormal subgroup) must also satisfy the second subgroup property (i.e., WNSCC-subgroup)
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Statement with symbols
Suppose is an abnormal subgroup of a group . Then, is a WNSCC-subgroup of : for any two normal subsets of with for some , we have .
(These definitions use the left action convention, but using the right action convention does not change any of the proofs).
Further information: Abnormal subgroup
A subgroup of a group is termed abnormal in if, for any , we have .
Further information: WNSCC-subgroup
A subgroup of a group is termed WNSCC in if, for any two normal subsets of such that there exists with , we have .
Given: A group , an abnormal subgroup . Two normal subsets of such that there exists with .
To prove: .
- and (here denotes the normalizer of the subset in ): This is a direct consequence of the fact that are normal subsets of .
- : Since , and conjugation by is an automorphism, we get , yielding .
- : This follows from the previous two steps.
- : Since is abnormal, . Combining this with the previous step yields .
- : Since , we have , so . Also, by assumption, . Combining these, we get .