# Minimal normal subgroup with order not dividing index is characteristic

From Groupprops

## Contents

## Statement

A minimal normal subgroup of a finite group whose order does not divide its index is characteristic.

## Related facts

## Facts used

- Automorphisms preserve the property of being normal and hence of being minimal normal.
- Normality is strongly intersection-closed
- Product formula

## Proof

**Given**: A finite group , a minimal normal subgroup of . The order of does not divide the index of in . An automorphism of .

**To prove**: .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is a minimal normal subgroup of . | Fact (1) | is an automorphism and is a minimal normal subgroup of . | Given-cum-fact direct. | |

2 | is a normal subgroup of . | Fact (2) | Step (1) | is normal and is also normal by Step (1). Thus, by Fact (2), is normal. | |

3 | Either is trivial, or . | Steps (1), (2) | By Step (2), is normal. Since is minimal normal, the normal subgroup is either trivial or equals . Similarly, since is minimal normal, the normal subgroup is either trivial or equals . Combining these, we get the conclusion. | ||

4 | If is trivial, then is a normal subgroup of of order . | Fact (3) | Step (1) | By Step (1), is normal. So is . Thus, their product is their join. If is trivial, the product has order by Fact (3), which simplifies to because . | |

5 | does not divide . | Fact (4) | does not divide . | By Lagrange's theorem, . dividing is equivalent to dividing . | |

6 | is nontrivial. | Fact (4) | Steps (4), (5) | If is trivial, then Step (4) gives us a subgroup of of order . By Fact (4), this would give that divides . However, by Step (5), does not divide . | |

7 | . | Steps (3) and (6) | Step-combination direct. |