# Permutable implies subnormal in finite

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Permutable subgroup (?)) must also satisfy the second subgroup property (i.e., Subnormal subgroup (?)). In other words, every permutable subgroup of finite group is a subnormal subgroup of finite group.

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## Contents

## Statement

Any permutable subgroup of a finite group is a subnormal subgroup. In particular, it is a permutable subnormal subgroup.

## Related facts

### Similar facts

- Conjugate-permutable implies subnormal in finite
- Permutable implies ascendant
- Permutable implies locally subnormal
- Permutable implies subnormal in finitely generated

### Generalizations of proof technique

- Max-equivalent to normal and intermediate subgroup condition implies subnormal in finite: Other examples of this proof technique are:

## Facts used

## Proof

**Given**: A finite group , a permutable subgroup of .

**To prove**: is subnormal in .

**Proof**: We prove this by induction on the order of .

If , we are done. Otherwise:

- Let be a maximal element among the proper permutable subgroups of containing . By fact (1), is normal in .
- By fact (2), is permutable in , so induction on the order yields that is subnormal in .
- Thus, is subnormal in which is normal in , so is subnormal in .