# Frattini's argument

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This article gives a proof/explanation of the equivalence of multiple definitions for the term automorph-conjugate subgroup

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## Contents

## Statement

### For automorph-conjugate subgroups

Let be a normal subgroup of and an automorph-conjugate subgroup of . Then:

where denotes the normalizer of in .

### For Sylow subgroups

Let be a normal subgroup of and a Sylow subgroup (?) of . Then:

where denotes the normalizer of in .

### For characteristic subgroups of Sylow subgroups

Let be a normal subgroup of , be a Sylow subgroup of , and be a characteristic subgroup of . In other words, is a Characteristic subgroup of Sylow subgroup (?) of . Then:

.

## Facts used

- Sylow implies automorph-conjugate
- Characteristic implies automorph-conjugate
- Automorph-conjugacy is transitive

## Proof

### Proof for automorph-conjugate subgroups

(*This proof uses the left action convention*)

**Given**: a normal subgroup of . an automorph-conjugate subgroup of .

**To prove**: .

**Proof**: Let . Consider . Since is normal, the map is an automorphism restricted to . Since is automorph-conjugate in , there exists such that .

Then, , and hence , with . thus, every element of can be expressed as the product of an element of and an element of , and we are done.

### Proof for Sylow subgroups

This follows from the statement for automorph-conjugate subgroups and fact (1).

### Proof for characteristic subgroups

By facts (1), (2) and (4), every characteristic subgroup of a Sylow subgroup is automorph-conjugate, so this statement again follows from the statement for automorph-conjugate subgroups.