# Isomorphism between linear groups when degree power map is bijective

From Groupprops

## Contents

## Statement

### Statement for an arbitrary field

Suppose is a field and is a natural number such that the map is a bijection from to itself. Then, the following are true:

- The general linear group is an internal direct product of these two subgroups: the special linear group and the center , which is the group of scalar matrices and is isomorphic to the multiplicative group .
- The composite of the inclusion of in and the quotient map from to the projective general linear group is an isomorphism.
- In particular, from (2), we get that .

### Statement for a finite field

Suppose is a prime power and is a natural number such that . Then the field of size satisfies the condition that the map is a bijection, and the following are true:

- The general linear group is an internal direct product of these two subgroups: the special linear group and the center , which is the group of scalar matrices and is isomorphic to the multiplicative group , which is a cyclic group of order (see multiplicative group of a finite field is cyclic).
- The composite of the inclusion of in and the quotient map from to the projective general linear group is an isomorphism.
- In particular, from (2), we get that .

## Particular cases

Congruence condition on for the isomorphisms to hold (basically, need ) | Initial examples of | |
---|---|---|

1 | all | |

2 | is even | : (all are isomorphic to symmetric group:S3) (all are isomorphic to alternating group:A5) (see projective special linear group:PSL(2,8)) |

3 | is 0 or -1 mod 3 | : (see projective special linear group:PSL(3,2) (see projective special linear group:PSL(3,3)) |

4 | is even | |

5 | is not 1 mod 5 |