# 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

## Contents

## 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 .

## Proof

### (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 .