# Characteristic implies automorph-conjugate

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) must also satisfy the second subgroup property (i.e., automorph-conjugate subgroup)

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## Statement

### Verbal statement

Any characteristic subgroup of a group is an automorph-conjugate subgroup.

### Symbolic statement

Let be a characteristic subgroup of . Then is an automorph-conjugate subgroup of .

### Property-theoretic statement

The subgroup property of being characteristic is *stronger than* the subgroup property of being automorph-connjugate.

## Definitions used

### Characteristic subgroup

`Further information: Characteristic subgroup`

A subgroup of a group is termed a characteristic subgroup if any automorphism of the group maps the subgroup to itself.

That is, is characteristic if for any automorphism of , .

Characteristicity can also be expressed using the relation implication expression:

Automorphic subgroups Equal subgroups

In other words, any subgroup obtained as where is an automorphism of , is equal to .

### Automorph-conjugate subgroup

`Further information: Automorph-conjugate subgroup`
A subgroup of a group is termed an **automorph-conjugate subgroup** if any automorphism of the group maps the subgroup to a conjugate subgroup.

That is is automorph-conjugate if for any automorphism of there exists such that (the latter is sometimes denoted ).

This property can also be expressed using the relation implication expression:

Automorphic subgroups Conjugate subgroups

In other words, any subgroup obtained as where is an automorphism of , is also a conjugate subgroup to .

## Proof

### Hands-on proof

*Given*: is a characteristic subgroup of ,

*To prove*: is an automorph-conjugate subgroup of . In other words, for any automorphism of , there exists , .

*Proof*: For any automorphism of , (by definition of characteristicity). Clearly is a conjugate subgroup to itself (say where is the identity element). Thus, is automorph-conjugate.

### Using relation implication expressions

This subgroup property implication can be proved using relation implication expressions for the subgroup properties

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The property of being a characteristic subgroup can be expressed as a relation implication:

Automorphic subgroups Equal subgroups

The property of being an automorph-conjugate subgroup can be expressed as a relation implication:

Automorphic subgroups Conjugate subgroups

Since the left side is the same for both properties, but the right side is stronger for characteristicity, we see that characteristic implies automorph-conjugate.

## Converse

The converse is *not* true. Counterexamples can be easily constructed by looking at Sylow subgroups which are not normal. This is because any Sylow subgroup must be an isomorph-conjugate subgroup, and hence an automorph-conjugate subgroup.