# 2-hypernormalized satisfies intermediate subgroup condition

From Groupprops

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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Get more facts about 2-hypernormalized subgroup |Get facts that use property satisfaction of 2-hypernormalized subgroup | Get facts that use property satisfaction of 2-hypernormalized subgroup|Get more facts about intermediate subgroup condition

## Contents

## Statement

Suppose is a 2-hypernormalized subgroup of a group . In other words, . Suppose is a subgroup of containing . Then, is 2-hypernormalized in : .

## Related facts

- Finitarily hypernormalized does not satisfy intermediate subgroup condition
- 2-hypernormalized does not satisfy transfer condition

## Facts used

## Proof

**Given**: , .

**To prove**: .

**Proof**:

- : 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.