# Element structure of special linear group:SL(2,9)

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This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,9).

View element structure of particular groups | View other specific information about special linear group:SL(2,9)

This article describes the element structure of special linear group:SL(2,9).

## Summary

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

order of the whole group (total number of elements) | 720 |

conjugacy class sizes | 1,1,40,40,40,40,72,72,72,72,90,90,90 in grouped form: 1 (2 times), 40 (4 times), 72 (4 times), 90 (3 times) maximum: 90, number of conjugacy classes: 13, lcm: 360 |

order statistics | 1 of order 1, 1 of order 2, 80 of order 3, 90 of order 4, 144 of order 5, 80 of order 6, 180 of order 8, 144 of order 10 maximum: 10, lcm (exponent of the whole group): 120 |

## Elements

### Order computation

The group has order 720. with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 720:

Family | Parameter values | Formula for order of a group in the family | Proof or justification of formula | Evaluation at parameter values | Full interpretation of conjugacy class structure |
---|---|---|---|---|---|

special linear group of degree two over a finite field of size | , i.e., field:F9, so the group is | , in factored form | See order formulas for linear groups of degree two, order formulas for linear groups, and special linear group of degree two | Factored version: |
#Interpretation as special linear group of degree two |

double cover of alternating group of degree | degree , so the group is | See double cover of alternating group, element structure of double cover of alternating group | #Interpretation as double cover of alternating group |

## Conjugacy class structure

### Interpretation as special linear group of degree two

`Further information: element structure of special linear group of degree two over a finite field`

Nature of conjugacy class | Eigenvalue pairs of all conjugacy classes | Characteristic polynomials of all conjugacy classes | Minimal polynomials of all conjugacy classes | Size of conjugacy class (generic odd ) | Size of conjugacy class () | Number of such conjugacy classes (generic odd ) | Number of such conjugacy classes () | Total number of elements (generic odd ) | Total number of elements () | Representative matrices (one per conjugacy class) |
---|---|---|---|---|---|---|---|---|---|---|

Scalar | or | or | or | 1 | 1 | 2 | 2 | 2 | 2 | and |

Not diagonal, Jordan block of size two | or | or | or | 40 | 4 | 4 | 160 | [SHOW MORE] where is a non-square in field:F9 | ||

Diagonalizable over field:F81, not over field:F9. Must necessarily have no repeated eigenvalues. | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
72 | 4 | 288 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| |||

Diagonalizable over field:F9 with distinct diagonal entries |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
90 | 3 | 270 | -- | |||

Total | NA | NA | NA | NA | NA | 13 | 720 | NA |

### Interpretation as double cover of alternating group

`Further information: element structure of double cover of alternating group`

is isomorphic to . Recall that we have the following rules to determine splitting and orders. The rules listed below are *only* for partitions that already correspond to even permutations, i.e., partitions that have an even number of even parts:

Hypothesis: does the partition have at least one even part? | Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd) | Conclusion: does the conjugacy class split from to in 2? | Conclusion: does the fiber in over a conjugacy class in split in 2? | Total number of conjugacy classes in corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both) | Number of these conjugacy classes where order of element = lcm of parts | Number of these conjugacy classes where order of element = twice the lcm of parts |
---|---|---|---|---|---|---|

No | No | Yes | Yes | 4 | 2 | 2 |

No | Yes | No | Yes | 2 | 1 | 1 |

Yes | No | No | Yes | 2 | 0 | 2 |

Yes | Yes | No | No | 1 | 0 | 1 |

Partition | Partition in grouped form | Does the partition have at least one even part? | Does the partition have a repeated part? | Conclusion: does the conjugacy class split from to in 2? | Conclusion: does the fiber in over a conjugacy class in split in 2? | Total number of conjugacy classes in corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both) | Size of each conjugacy class | Size formula (we take the size formula in , multiply by 2, and divide by the number (1,2, or 4) two columns preceding | Total number of elements (= twice the size of the -conjugacy class) | Element orders | Formula for element orders |
---|---|---|---|---|---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 + 1 | 1 (6 times) | No | Yes | No | Yes | 2 | 1 | 2 | 1 (1 class), 2 (1 class) | (1 class) (1 class) | |

3 + 1 + 1 + 1 | 3 (1 time), 1 (3 times) | No | Yes | No | Yes | 2 | 40 | 80 | 3 (1 class), 6 (1 class) | (1 class) (1 class) | |

2 + 2 + 1 + 1 | 2 (2 times), 1 (2 times) | Yes | Yes | No | No | 1 | 90 | 90 | 4 (1 class) | ||

5 + 1 | 5 (1 time), 1 (1 time) | No | No | Yes | Yes | 4 | 72 | 288 | 5 (2 classes), 10 (2 classes) | (1 class) (1 class) | |

4 + 2 | 4 (1 time), 2 (1 time) | Yes | No | No | Yes | 2 | 90 | 180 | 8 (2 classes) | (2 classes) | |

3 + 3 | 3 (2 times) | No | Yes | No | Yes | 2 | 40 | 80 | 3 (1 class), 6 (1 class) | (1 class) (1 class) | |

Total | -- | -- | -- | -- | -- | 13 | -- | -- | 720 | -- | -- |

## Conjugacy class structure: additional information

### Number of conjugacy classes

The group has 13 conjugacy classes. The number can be computed in a number of ways:

Family | Parameter values | Formula for number of conjugacy classes of a group in the family | Proof or justification of formula | Evaluation at parameter values | Full interpretation of conjugacy class structure |
---|---|---|---|---|---|

special linear group of degree two over a finite field of size | , i.e., field:F9 | Case odd: Case even: |
element structure of special linear group of degree two over a finite field; see also [[number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size | Since 9 is odd, we use the odd case formula, and get | #Interpretation as special linear group of degree two |

double cover of alternating group | , i.e., the group is double cover of alternating group:A6 | (number of unordered integer partitions of ) + 3(number of partitions of into distinct odd parts) - (number of partitions of with a positive even number of even parts and with at least one repeated part) | See element structure of double cover of alternating group, splitting criterion for conjugacy classes in double cover of alternating group | For , the three numbers to calculate are respectively 11,1,1. So, we get . | #Interpretation as double cover of alternating group |