# Subgroup structure of special linear group:SL(2,3)

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: special linear group:SL(2,3).

View subgroup structure of particular groups | View other specific information about special linear group:SL(2,3)

This article gives information on the subgroup structure of special linear group:SL(2,3), which is the special linear group of degree two over field:F3. Note that this field has three elements and we have .

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)

Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)

Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order

Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugateMINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

### Quick summary

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

number of subgroups | 15 |

number of conjugacy classes of subgroups | 7 |

number of automorphism classes of subgroups | 7 |

isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers | 2-Sylow: quaternion group (order 8) as Q8 in SL(2,3) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 1 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 4 |

Hall subgroups | The order has only two prime divisors, so no possibility of Hall subgroups other than trivial subgroup, whole group, and Sylow subgroups |

maximal subgroups | There are maximal subgroups of orders 6 (Z6 in SL(2,3)) and 8 (2-Sylow subgroup of special linear group:SL(2,3)) |

normal subgroups | There are two proper nontrivial normal subgroups: center of special linear group:SL(2,3) and 2-Sylow subgroup of special linear group:SL(2,3) |

### Table classifying subgroups up to automorphism

Note that, in the matrices, -1 can be written as 2 since elements are taken modulo 3.

Automorphism class of subgroups | Representative subgroup | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) | Note |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 24 | 1 | 1 | 1 | special linear group:SL(2,3) | 1 | trivial | |

center of special linear group:SL(2,3) | cyclic group:Z2 | 2 | 12 | 1 | 1 | 1 | alternating group:A4 | 1 | ||

Z4 in SL(2,3) | cyclic group:Z4 | 4 | 6 | 1 | 3 | 3 | -- | 2 | ||

2-Sylow subgroup of special linear group:SL(2,3) | quaternion group | 8 | 3 | 1 | 1 | 1 | cyclic group:Z3 | 1 | 2-Sylow | |

Z3 in SL(2,3) | cyclic group:Z3 | 3 | 8 | 1 | 4 | 4 | -- | -- | 3-Sylow | |

Z6 in SL(2,3) | cyclic group:Z6 | 6 | 4 | 1 | 4 | 4 | -- | -- | 3-Sylow normalizer | |

whole group | all elements | special linear group:SL(2,3) | 24 | 1 | 1 | 1 | 1 | trivial group | 0 | |

Total (7 rows) | -- | -- | -- | -- | 7 | -- | 15 | -- | -- | -- |

### Table classifying isomorphism types of subgroups

Group name | GAP ID | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | 1 | |

Cyclic group:Z2 | 1 | 1 | 1 | 1 | 1 | |

Cyclic group:Z3 | 4 | 1 | 1 | 0 | 0 | |

Cyclic group:Z4 | 3 | 1 | 1 | 0 | 0 | |

Cyclic group:Z6 | 4 | 1 | 1 | 0 | 0 | |

Quaternion group | 1 | 1 | 1 | 1 | 1 | |

Special linear group:SL(2,3) | 1 | 1 | 1 | 1 | 1 | |

Total | -- | 15 | 7 | 7 | 4 | 4 |

### Table listing number of subgroups by order

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 1 | 1 | 1 | 1 |

3 | 4 | 1 | 0 | 0 |

4 | 3 | 1 | 0 | 0 |

6 | 4 | 1 | 0 | 0 |

8 | 1 | 1 | 1 | 1 |

24 | 1 | 1 | 1 | 1 |

Total | 15 | 7 | 4 | 4 |

## Defining functions

### Subgroup-defining functions and associated quotient-defining functions

## Sylow subgroups

Compare and contrast with subgroup structure of special linear group of degree two over a finite field#Sylow subgroups

We are considering the group with a prime power, . The prime is the *characteristic prime*.

### Sylow subgroups for the prime 3

The prime 3 is the *characteristic prime* , so we compare with the general information on -Sylow subgroups of .

Item | Value for , generic | Value for (so ) |
---|---|---|

order of -Sylow subgroup | 3 | |

index of -Sylow subgroup | 8 | |

explicit description of one of the -Sylow subgroups | unitriangular matrix group of degree two: | . See 3-Sylow subgroup of special linear group:SL(2,3) |

isomorphism class of -Sylow subgroup | additive group of the field , which is an elementary abelian group of order | cyclic group:Z3 |

explicit description of -Sylow normalizer | Borel subgroup of degree two: | See Z6 in SL(2,3) |

isomorphism class of -Sylow normalizer | It is the external semidirect product of by the multiplicative group of where the latter acts on the former via the multiplication action of the square of the acting element.For (so ), it is isomorphic to the general affine group of degree one . For , it is cyclic group:Z6 and for , it is dicyclic group:Dic20. |
cyclic group:Z6 |

order of -Sylow normalizer | 6 | |

-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) | 4 |

### Sylow subgroups for the prime 2

We are in the subcase where ( being the prime for which we are taking Sylow subgroups) and .

Item | Value for | Value for (our case) |
---|---|---|

order of 2-Sylow subgroup | 8 | 8 |

index of 2-Sylow subgroup | 3 | |

explicit description of one of the 2-Sylow subgroups | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
See 2-Sylow subgroup of special linear group:SL(2,3) |

isomorphism class of 2-Sylow subgroup | quaternion group | quaternion group |

explicit description of 2-Sylow normalizer | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
whole group |

isomorphism class of 2-Sylow normalizer | special linear group:SL(2,3) | special linear group:SL(2,3) |

order of 2-Sylow normalizer | 24 | 24 |

2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer | 1 |