# Subgroup structure of projective general linear group of degree two over a finite field

From Groupprops

This article gives specific information, namely, subgroup structure, about a family of groups, namely: projective general linear group of degree two.

View subgroup structure of group families | View other specific information about projective general linear group of degree two

## Particular cases

## Sylow subgroups

We consider the group over the field of elements. is a prime power of the form where is a prime number and is a positive integer. is hence also the characteristic of . We call the *characteristic prime*.

### Sylow subgroups for the characteristic prime

Item | Value |
---|---|

order of -Sylow subgroup | |

index of -Sylow subgroup | |

explicit description of one of the -Sylow subgroups | image of unitriangular matrix group of degree two: |

isomorphism class of -Sylow subgroup | additive group of , which is an elementary abelian group of order , i.e., a direct product of copies of the cyclic group of order |

explicit description of -Sylow normalizer | image of Borel subgroup of degree two in : |

isomorphism class of -Sylow normalizer | general affine group of degree one |

order of -Sylow normalizer | |

-Sylow number (i.e., number of -Sylow subgroups) = index of -Sylow normalizer | (congruent to 1 mod p, as expected from the congruence condition on Sylow numbers) |

### Sylow subgroups for other primes: cases and summary

For any prime , the -Sylow subgroup is nontrivial iff . If , then it does not divide , so we get that which means that either or . Further, if , exactly one of these cases can occur. For , we make cases based on the residue of mod 4:

Case on and | Isomorphism type of -Sylow subgroup | Isomorphism type of -Sylow normalizer | Order of -Sylow normalizer | -Sylow number = index of -Sylow normalizer |
---|---|---|---|---|

is an odd prime dividing , | cyclic group | dihedral group | ||

is an odd prime dividing , | cyclic group | dihedral group | ||

is an odd prime dividing , | cyclic group | dihedral group | ||

is an odd prime dividing , | cyclic group | dihedral group | ||

and | dihedral group | dihedral group | largest power of 2 dividing the order = twice the largest power of 2 dividing | largest odd number dividing the order |

and | dihedral group | dihedral group | largest power of 2 dividing the order = twice the largest power of 2 dividing | largest odd number dividing the order |