# Sylow implies intermediately isomorph-conjugate

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., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Intermediately isomorph-conjugate subgroup (?)). In other words, every Sylow subgroup of finite group is a intermediately isomorph-conjugate subgroup of finite group.

View all subgroup property implications in finite groups View all subgroup property non-implications in finite groups View all subgroup property implications View all subgroup property non-implications

## Statement

A Sylow subgroup of a finite group is intermediately isomorph-conjugate: it is isomorph-conjugate in every intermediate subgroup.

## Facts used

- Sylow satisfies intermediate subgroup condition: A Sylow subgroup of a group is Sylow in every intermediate subgroup.
- Sylow implies isomorph-conjugate

## Proof

The proof follows directly by combining facts (1) and (2).