# Subgroup structure of projective special 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 special linear group of degree two.

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

## Particular cases

Group | order of the group (= if odd, if even) | subgroup structure page | ||
---|---|---|---|---|

symmetric group:S3 | 2 | 2 | 6 | subgroup structure of symmetric group:S3 |

alternating group:A4 | 3 | 3 | 12 | subgroup structure of alternating group:A4 |

alternating group:A5 | 2 | 4 | 60 | subgroup structure of alternating group:A5 |

alternating group:A5 | 5 | 5 | 60 | subgroup structure of alternating group:A5 |

projective special linear group:PSL(3,2) | 7 | 7 | 168 | subgroup structure of projective special linear group:PSL(3,2) |

projective special linear group:PSL(2,8) | 2 | 8 | 504 | subgroup structure of projective special linear group:PSL(2,8) |

alternating group:A6 | 3 | 9 | 360 | subgroup structure of alternating group:A6 |

## 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*.

Note that when ( even), the order of the group is . When ( odd), the order of the group is .

### Sylow subgroups for the characteristic prime

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

order of -Sylow subgroup | |

index of -Sylow subgroup | Case : Case : |

explicit description of one of the -Sylow subgroups | image mod the center of unitriangular matrix group of degree two: Note that this subgroup intersects the center trivially, so there are no identifications when taking the quotient by the center |

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 mod the center of Borel subgroup of degree two: . Explicitly, we identify the matrix up to sign on the whole matrix. |

isomorphism class of -Sylow normalizer | It is the external semidirect product of by the quotient where the latter acts on the former via the multiplication action of the square of the acting element. Note that the action is well defined because the square of acts trivially.Equivalently, it is the obvious subgroup of the general affine group of degree one . For (so ), it is isomorphic to the general affine group of degree one . For , it is cyclic group:Z3 and for , it is dihedral group:D10. |

order of -Sylow normalizer | Case , i.e., even: Case , i.e., odd: |

-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 8. The summary of cases is below and more details are in later sections.

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 | dicyclic 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 | Klein four-group | alternating group:A4 | 12 | |

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 | Klein four-group | alternating group:A4 | 12 |