2-hypernormalized satisfies intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-hypernormalized subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Suppose is a 2-hypernormalized subgroup of a group . In other words, . Suppose is a subgroup of containing . Then, is 2-hypernormalized in : .
- Finitarily hypernormalized does not satisfy intermediate subgroup condition
- 2-hypernormalized does not satisfy transfer condition
Given: , .
To prove: .
- : This follows directly from the definition of normalizer.
- is normal in : By assumption, is normal in . Fact (1) yields that is normal in .
- : This follows directly from the previous step.