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

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

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

This article gives detailed information about the element structure of special linear group:SL(2,3).

See also element structure of special linear group of degree two over a finite field.

## Summary

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

order of the whole group (total number of elements) | 24 |

conjugacy class sizes | 1,1,4,4,4,4,6 grouped form: 1 (2 times), 4 (4 times), 6 (1 time) maximum: 6, number of conjugacy classes: 7, lcm: 12 |

order statistics | 1 of order 1, 1 of order 2, 8 of order 3, 6 of order 4, 8 of order 6 maximum: 6, lcm (exponent of the whole group): 12 |

## Elements

### Order computation

The group has order 24. with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 24:

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:F3, 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 | ||

binary von Dyck group with parameters | (note that the order of the parameters is irrelevant, though we usually arrange them in ascending or descending order depending on the convention being followed). | See element structure of binary von Dyck groups | #Interpretation as binary von Dyck group |

## Conjugacy and automorphism class structure

### Conjugacy classes

Note that since we are over field:F3, , so all the s below can be rewritten as s.

COMBINATORIAL BREAKDOWN TABLE: The table below breaks down a collection into various classes or types and provides information on the counts for each type. For some of the columns, totals provide a sanity check that all elements or classes have been accounted for. In this case, the table gives information on conjugacy class structure.

Conjugacy class representative | Conjugacy class size | List of all elements of conjugacy class | Order of elements in conjugacy class |
---|---|---|---|

1 | 1 | ||

1 | 2 | ||

4 | [SHOW MORE] | 3 | |

4 | [SHOW MORE] | 3 | |

4 | [SHOW MORE] | 6 | |

4 | [SHOW MORE] | 6 | |

6 | [SHOW MORE] | 4 | |

Total | 24 (order of the group) | -- | -- |

### Automorphism classes

COMBINATORIAL BREAKDOWN TABLE: The table below breaks down a collection into various classes or types and provides information on the counts for each type. For some of the columns, totals provide a sanity check that all elements or classes have been accounted for. In this case, the table gives information on automorphism class structure.

Below are the orbits under the action of the automorphism group, i.e., the automorphism classes of elements of the group.

List of representatives for each conjugacy class in the automorphism class | Number of conjugacy classes in the automorphism class | Size of each conjugacy class | Automorphism class size | Order of elements in conjugacy class |
---|---|---|---|---|

1 | 1 | 1 | 1 | |

1 | 1 | 1 | 2 | |

, | 2 | 4 | 8 | 3 |

, | 2 | 4 | 8 | 6 |

1 | 6 | 6 | 4 | |

Total | 7 (number of conjugacy classes) | -- | 24 (order of the group) | -- |

### Interpretation as special linear group of degree two

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

COMBINATORIAL BREAKDOWN TABLE: The table below breaks down a collection into various classes or types and provides information on the counts for each type. For some of the columns, totals provide a sanity check that all elements or classes have been accounted for. In this case, the table gives information on conjugacy class structure.

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 | 4 | 4 | 4 | 16 | [SHOW MORE] | ||

Diagonalizable over field:F9, not over field:F3. Must necessarily have no repeated eigenvalues. | pair of square roots of in field:F9 | 6 | 1 | 6 | ||||||

Diagonalizable over field:F3 with distinct diagonal entries |
-- | -- | -- | 12 | 0 | 0 | -- | |||

Total | NA | NA | NA | NA | NA | 7 | 24 | 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 (4 times) | No | Yes | No | Yes | 2 | 1 | 2 | 1 (1 class), 2 (1 class) | (1 class) (1 class) | |

2 + 2 | 2 (2 times) | Yes | Yes | No | No | 1 | 6 | 6 | 4 | (1 class) | |

3 + 1 | 3 (1 time), 1 (1 time) | No | No | Yes | Yes | 4 | 4 | 16 | 3 (2 classes) 6 (2 classes) |
(2 classes) (2 classes) | |

Total | -- | -- | -- | -- | -- | 7 | -- | -- | 24 | -- | -- |

## Conjugacy class structure: additional information

### Number of conjugacy classes

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., the group | , i.e., field:F3, so the group is . | for odd for a power of 2 |
See element structure of special linear group of degree two over a finite field, number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size | #Interpretation as special linear group of degree two | |

double cover of alternating group of degree | , i.e., the group | (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 | The three numbers compute to 5, 1, and 1 respectively, so we get | #Interpretation as double cover of alternating group |

binary von Dyck group with parameters | (note that the order of the parameters is irrelevant, though we usually arrange them in ascending or descending order depending on the convention being followed). | See element structure of binary von Dyck groups | #Interpretation as binary von Dyck group |