# Order of periodic element of general linear group over integers is bounded

From Groupprops

This article gives the statement, and possibly proof, of a particular group or type of group (namely, General linear group over integers (?)) satisfying a particular group property (namely, Group in which all elements of finite order have a common bound on order (?)).

## Contents

## Statement

Suppose is a natural number. Then, the order of any element of the General linear group (?) having finite order is *bounded* by a function of .

Explicitly, if there exists an element of order , then we can write where denotes the Euler totient function, all , and the lcm of equals .

This gives a bounding function of the form: the maximum possible order of a matrix is at most . In practice, the maximum possible order is much lower.

## Particular cases

Value of | Possible orders of periodic elements in | Maximum possible order | lcm of possible orders |
---|---|---|---|

1 | 1,2 | 2 | 2 |

2 | 1,2,3,4,6 | 6 | 12 |

3 | 1,2,3,4,6 | 6 | 12 |

4 | 1,2,3,4,5,6,8,10,12 | 12 | 120 |

5 | 1,2,3,4,5,6,8,10,12 | 12 | 120 |

6 | 1,2,3,4,5,6,7,8,9,10,12,14,15,20,24,30 | 30 | 2520 |

## Proof

### Proof of the Euler totient function partition condition

**Given**: An element of order .

**To prove**: There exists a partition where the lcm of is and is the Euler totient function.

**Proof**:

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

1 | satisfies the polynomial , so the minimal polynomial of divides . | has order | |||

2 | The minimal polynomial of is a product of cyclotomic polynomials , where the lcm of is . | , are irreducible | has order . | Step (1) | By Step (1) and the factoring of we conclude that the minimal polynomial must be a product of for a subset of the set of divisors of . Call these divisors . They all divide . If the lcm is smaller than , then the order of is strictly smaller than , a contradiction. Hence the lcm must be exactly . |

3 | The characteristic polynomial is of the form where are all positive integers, and the lcm of is . | minimal polynomial divides characteristic polynomial | Step (2) | Follows from fact and Step (2). | |

4 | , where are all positive integers, and the lcm of is . | degree of cyclotomic polynomial is , degree of characteristic polynomial of matrix is | Step (3) | Direct from Step (3), compute degree of characteristic polynomial in two different ways. |

### Proof of bounding

**Given**: , where are all positive integers, and the lcm of is .

**To prove**: .

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

1 | Each is bounded by . | ||||

2 | Each . | for any natural number . | Step (1) | ||

3 | |||||

4 | , defined as the lcm of is at most . | ||||

5 | Steps (3), (4), (5) | Step-combination direct. |