# Difference between revisions of "Element structure of special linear group:SL(2,5)"

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− | ! 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 <matH>S_n</math> to <math>A_n</math> in 2? !! Conclusion: does the fiber in <math>2 \cdot A_n</math> over a conjugacy class in <math>A_n</math> split in 2? !! Total number of conjugacy classes in <matH>2 \cdot A_n</math> 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 <matH>S_n</math>, multiply by 2, and divide by the number (1,2, or 4) two columns preceding !! Total number of elements (= twice the size of the <math>S_n</math>-conjugacy class) !! Element orders !! Formula for element orders | + | ! 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 <matH>S_n</math> to <math>A_n</math> in 2? !! Conclusion: does the fiber in <math>2 \cdot A_n</math> over a conjugacy class in <math>A_n</math> split in 2? !! Total number of conjugacy classes in <matH>2 \cdot A_n</math> 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 <matH>S_n</math>, multiply by 2, and divide by the number (1,2, or 4) two columns preceding) !! Total number of elements (= twice the size of the <math>S_n</math>-conjugacy class) !! Element orders !! Formula for element orders |

|- | |- | ||

| 1 + 1 + 1 + 1 + 1 || 1 (5 times) || No || Yes || No || Yes || 2 || 1 || <math>\frac{2}{2} \frac{5!}{(1)^5(5!)}</math> || 2 || 1 (1 class), 2 (1 class) || <math>\operatorname{lcm} \{ 1 \} </math> (1 class)<br><math>2\operatorname{lcm} \{ 1 \}</math> (1 class) | | 1 + 1 + 1 + 1 + 1 || 1 (5 times) || No || Yes || No || Yes || 2 || 1 || <math>\frac{2}{2} \frac{5!}{(1)^5(5!)}</math> || 2 || 1 (1 class), 2 (1 class) || <math>\operatorname{lcm} \{ 1 \} </math> (1 class)<br><math>2\operatorname{lcm} \{ 1 \}</math> (1 class) |

## Latest revision as of 07:06, 13 August 2016

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,5).

View element structure of particular groups | View other specific information about special linear group:SL(2,5)

This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

## Summary

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

order of the whole group (total number of elements) | 120 |

conjugacy class sizes | 1,1,12,12,12,12,20,20,30 in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time) maximum: 30, number of conjugacy classes: 9, lcm: 60 |

order statistics | 1 of order 1, 1 of order 2, 20 of order 3, 30 of order 4, 24 of order 5, 20 of order 6, 24 of order 10 maximum: 10, lcm (exponent of the whole group): 60 |

## Elements

### Order computation

The group has order 120. with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 120:

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:F5, 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 class structure

### Conjugacy classes

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Interpretation as special linear group of degree two

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

In the table below, . Note that the information is presented for generic odd and then computed numerically for .

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 | 12 | 4 | 4 | 48 | [SHOW MORE] | ||

Diagonalizable over (in our case field:F25), not over (in our case, field:F5). Must necessarily have no repeated eigenvalues. | For : and , where is interpreted a an element of field:F25 that squares to 3 | For : , | For : , | 20 | 2 | 40 | , | |||

Diagonalizable over , i.e., field:F5, with distinct diagonal entries |
For : | For : | For : | 30 | 1 | 30 | ||||

Total | NA | NA | NA | NA | NA | 9 | 120 | 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 (5 times) | No | Yes | No | Yes | 2 | 1 | 2 | 1 (1 class), 2 (1 class) | (1 class) (1 class) | |

2 + 2 + 1 | 2 (2 times), 1 (1 time) | Yes | Yes | No | No | 1 | 30 | 30 | 4 | (1 class) | |

3 + 1 + 1 | 3 (1 time), 1 (2 times) | No | Yes | No | Yes | 2 | 20 | 40 | 3 (1 class) 6 (1 class) |
(1 class) (1 class) | |

5 | 5 (1 time) | No | No | Yes | Yes | 4 | 12 | 48 | 5 (2 classes), 10 (2 classes) | (2 classes) (2 classes) | |

Total | -- | -- | -- | -- | -- | 9 | -- | -- | 120 | -- | -- |

## Conjugacy class structure: additional information

### Number of conjugacy classes

The group has 9 conjugacy classes. This number can be computed in a variety 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:F5 | 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 5 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:A5 | (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 7,1,1. So, we get . | #Interpretation as double cover of alternating group |

binary von Dyck group with parameters satisfying | #Interpretation as binary von Dyck group |