# Commensurator of subgroup is subgroup

From Groupprops

## Contents

## Statement

Suppose is a subgroup of a group . Consider the commensurator of in , defined as the set of all such that is a subgroup of finite index in both and , i.e., and are Commensurable subgroups (?). Then, is a subgroup of .

## Facts used

- Group acts as automorphisms by conjugation
- Index satisfies transfer inequality
- Index is multiplicative

## Proof

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**Given**: A group , a subgroup of . is the set of all such that has finite index in both and .

**To prove**: is a subgroup of .

**Proof**:

### Proof for identity element

Let denote the identity element of . We have , so the intersection also equals . This has index in both and , which is finite.

### Proof for inverses

**Additional given**: . In other words, has finite index in both and .

**To prove**: , i.e., has finite index in both and .

**Proof**

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

1 | Consider the map given by . This is an inner automorphism of | Fact (1) | -- | -- | |

2 | preserves intersections and index of subgroups | Follows from definition of automorphism | Step (1) | ||

3 | and . | Step (1) | |||

4 | . | Steps (2), (3) | |||

5 | has finite index in both and . | , the commensurator of . | |||

6 | has finite index in both and . | Steps (2), (3), (4), (5) | [SHOW MORE] | ||

7 | is in . | definition of | Step (6) | Step-definition direct. |

### Proof for products

**Additional given**:

**To prove**: , i.e., has finite index in both and in .

**Proof**:

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

1 | Consider the map given by . Then, is an inner automorphism and in particular an automorphism of . | Fact (1) | |||

2 | preserves intersections and index of subgroups | follows from definition of automorphism | Step (1) | ||

3 | and . | Step (1) | |||

4 | . | Steps (2), (3) | |||

5 | has finite index in both and . | , definition of | Given-direct | ||

6 | has finite index in (i) and (ii) | Steps (2), (3), (4), (5) | [SHOW MORE] | ||

7 | has finite index in (i) and (ii) . | Fact (2) | Step (6) | [SHOW MORE] | |

8 | has finite index in (i) and in (ii) | , definition of | Given-direct | ||

9 | has finite index in . | Fact (3) | Steps (7)(i), (8)(i) | [SHOW MORE] | |

10 | has finite index in | Fact (3) | Step (9) | [SHOW MORE] | |

11 | has finite index in (i) and (ii) | Fact (2) | Step (8) | [SHOW MORE] | |

12 | has finite index in | Fact (3) | Steps (6)(ii), (11)(ii) | [SHOW MORE] | |

13 | has finite index in | Fact (3) | Step (12) | [SHOW MORE] | |

14 | , i.e., has finite index in both and . | Steps (10), (13) | Step-combination direct. |