Equivalence of definitions of weakly abnormal subgroup
This article gives a proof/explanation of the equivalence of multiple definitions for the term weakly abnormal subgroup
View a complete list of pages giving proofs of equivalence of definitions
The definitions that we have to prove as equivalent
Here are two equivalent definitions of weakly abnormal for a subgroup of a group :
- For any , let be the closure in of under the action by conjugation of the cyclic group generated by . Then, .
- If , then is a Self-normalizing subgroup (?) of .
- If , then is a Contranormal subgroup (?) of .
(1) implies (2)
Given: such that for all . .
To prove: .
Proof: Suppose . Then, , so . In particular, , so . Thus, . is tautological, so .
(2) implies (1)
Given: such that, for any intermediate subgroup , .
To prove: for any .
Proof: Let . Then, by definition of , both and map to within itself, so conjugation by gives an automorphism of . Thus, . By assumption, , so , yielding , as desired.
(2) implies (3)
Given: A subgroup of with the property that for every subgroup of containing .
To prove: If , is contranormal in .
Proof: Let be the normal closure of in . Since is normal in , . Since , , so , forcing . Thus, is contranormal in .
(3) implies (2)
Given: A subgroup of with the property that if , is contranormal in .
To prove: for any subgroup of containing .
Proof: Let . Then . By assumption, is contranormal in , and is normal in , forcing . Thus, we get .