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

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

− | + | ==Summary== | |

+ | <section begin="summary"/> | ||

+ | {| class="sortable" border="1" | ||

+ | ! Item !! Value | ||

+ | |- | ||

+ | | [[order of a group|order]] of the whole group (total number of elements) || 120 | ||

+ | |- | ||

+ | | [[conjugacy class size statistics of a finite group|conjugacy class sizes]] || 1,1,12,12,12,12,20,20,30<br>in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time)<br>[[maximum conjugacy class size|maximum]]: 30, [[number of conjugacy classes]]: 9, [[lcm of conjugacy class sizes|lcm]]: 60 | ||

+ | |- | ||

+ | | [[order statistics of a finite group|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<br>[[maximum of element orders|maximum]]: 10, [[exponent of a group|lcm (exponent of the whole group)]]: 60 | ||

+ | |} | ||

+ | <section end="summary"/> | ||

+ | |||

+ | ==Elements== | ||

+ | |||

+ | ===Order computation=== | ||

+ | |||

+ | The group <math>SL(2,5)</math> has order 120. with prime factorization <math>120 = 2^3 \cdot 3^1 \cdot 5^1 = 8 \cdot 3 \cdot 5</math>. Below are listed various methods that can be used to compute the order, all of which should give the answer 120: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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 [[special linear group of degree two|degree two]] over a [[finite field]] of size <math>q</math> || <math>q = 5</math>, i.e., [[field:F5]], so the group is <math>SL(2,5)</math> || <math>q^3 - q</math>, in factored form <math>q(q - 1)(q + 1)</math> || See [[order formulas for linear groups of degree two]], [[order formulas for linear groups]], and [[special linear group of degree two]] || <math>5^3 - 5 = 120</math><br>Factored version: <math>5(5 - 1)(5 + 1) = 5(4)(6) = 120</math> || [[#Interpretation as special linear group of degree two]] | ||

+ | |- | ||

+ | | [[double cover of alternating group]] <math>2 \cdot A_n</math> of degree <math>n</math> || degree <math>n = 5</math>, so the group is <math>2 \cdot A_5</math> || <math>n!</math> || See [[double cover of alternating group]], [[element structure of double cover of alternating group]] || <math>5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math> || [[#Interpretation as double cover of alternating group]] | ||

+ | |- | ||

+ | | [[binary von Dyck group]] with parameters <math>(p,q,r)</math> || <math>(p,q,r) = (5,3,2)</math> (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). || <math>\frac{4}{1/p + 1/q + 1/r - 1}</math> || See [[element structure of binary von Dyck groups]] || <math>\frac{4}{1/5 + 1/3 + 1/2 - 1} = \frac{4}{1/30} = 120</math> || [[#Interpretation as binary von Dyck group]] | ||

+ | |} | ||

==Conjugacy class structure== | ==Conjugacy class structure== | ||

Line 14: | Line 41: | ||

{{fillin}} | {{fillin}} | ||

− | === | + | ===Interpretation as special linear group of degree two=== |

{{further|[[element structure of special linear group of degree two over a finite field]]}} | {{further|[[element structure of special linear group of degree two over a finite field]]}} | ||

+ | In the table below, <math>q = 5</math>. Note that the information is presented for generic odd <math>q</math> and then computed numerically for <math>q = 5</math>. | ||

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

! 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 <math>q</math>) !! Size of conjugacy class (<math>q = 5</math>) !! Number of such conjugacy classes (generic odd <math>q</math>) !! Number of such conjugacy classes (<math>q = 5</math>) !! Total number of elements (generic odd <math>q</math>) !! Total number of elements (<math>q = 5</math>) !! Representative matrices (one per conjugacy class) | ! 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 <math>q</math>) !! Size of conjugacy class (<math>q = 5</math>) !! Number of such conjugacy classes (generic odd <math>q</math>) !! Number of such conjugacy classes (<math>q = 5</math>) !! Total number of elements (generic odd <math>q</math>) !! Total number of elements (<math>q = 5</math>) !! Representative matrices (one per conjugacy class) | ||

Line 25: | Line 53: | ||

| Not diagonal, Jordan block of size two || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math> || <math>(q^2 - 1)/2</math> || 12 || 4 || 4 || <math>2(q^2 - 1)</math> || 48 || <toggledisplay><math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}</math></toggledisplay> | | Not diagonal, Jordan block of size two || <math>\{ 1, 1 \}</math> or <math>\{ -1,-1\}</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math> || <math>x^2 - 2x + 1</math> or <math>x^2 + 2x + 1</math> || <math>(q^2 - 1)/2</math> || 12 || 4 || 4 || <math>2(q^2 - 1)</math> || 48 || <toggledisplay><math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}</math>, <math>\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}</math></toggledisplay> | ||

|- | |- | ||

− | | Diagonalizable over [[field:F25]], not over [[field:F5]]. Must necessarily have no repeated eigenvalues. || <math>\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}</math> and <math>\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}</math>, where <math>\sqrt{3}</math> is interpreted a an element of [[field:F25]] that squares to 3 || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>q(q - 1)</math> || 20 || <math>(q - 1)/2</math> || 2 || <math>q(q - 1)^2/2</math> || 40 || <math>\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}</math>, <math>\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}</math> | + | | Diagonalizable over <math>\mathbb{F}_{q^2}</math> (in our case [[field:F25]]), not over <math>\mathbb{F}_q</math> (in our case, [[field:F5]]). Must necessarily have no repeated eigenvalues. || For <math>q = 5</math>: <math>\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}</math> and <math>\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}</math>, where <math>\sqrt{3}</math> is interpreted a an element of [[field:F25]] that squares to 3 || For <math>q = 5</math>: <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || For <matH>q = 5</math>: <math>x^2 - x + 1</math>, <math>x^2 + x + 1</math> || <math>q(q - 1)</math> || 20 || <math>(q - 1)/2</math> || 2 || <math>q(q - 1)^2/2</math> || 40 || <math>\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}</math>, <math>\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}</math> |

+ | |- | ||

+ | | Diagonalizable over <math>\mathbb{F}_q</math>, i.e., [[field:F5]], with ''distinct'' diagonal entries || For <math>q = 5</math>: <math>\{ 2,3 \}</math> || For <matH>q = 5</math>: <math>x^2 + 1</math> || For <matH>q = 5</math>: <math>x^2 + 1</math> || <math>q(q+1)</math> || 30 || <math>(q - 3)/2</math> || 1 || <math>q(q+1)(q-3)/2</math> ||30 || <math>\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}</math> | ||

+ | |- | ||

+ | ! Total || NA || NA || NA || NA || NA || <math>q + 4</math> || 9 || <math>q^3 - q</math> || 120 || NA | ||

+ | |} | ||

+ | |||

+ | ===Interpretation as double cover of alternating group=== | ||

+ | |||

+ | {{further|[[element structure of double cover of alternating group]]}} | ||

+ | |||

+ | <matH>SL(2,5)</math> is isomorphic to <math>2 \cdot A_n,n = 5</math>. 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: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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 <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) !! 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 | ||

+ | |} | ||

+ | |||

+ | {| class="sortable" border="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 <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) | ||

+ | |- | ||

+ | | 2 + 2 + 1 || 2 (2 times), 1 (1 time) || Yes || Yes || No || No || 1 || 30 || <math>\frac{2}{1} \frac{5!}{(2)^2(2!)(1)}</math>|| 30 || 4 || <math>2\operatorname{lcm} \{ 2,1 \}</math> (1 class) | ||

+ | |- | ||

+ | | 3 + 1 + 1 || 3 (1 time), 1 (2 times) || No || Yes || No || Yes || 2 || 20 || <math>\! \frac{2}{2} \frac{5!}{(3)(1)^2(2!)}</math> || 40 || 3 (1 class)<br>6 (1 class) || <math>\operatorname{lcm} \{ 3,1 \}</math> (1 class)<br><math>2\operatorname{lcm} \{ 3,1 \}</math> (1 class) | ||

+ | |- | ||

+ | | 5 || 5 (1 time) || No || No || Yes || Yes || 4 || 12 || <math>\frac{2}{4} \frac{5!}{5}</math> || 48 || 5 (2 classes), 10 (2 classes) || <math>\operatorname{lcm} \{ 5 \}</math> (2 classes)<br><math>2 \operatorname{lcm} \{ 5 \}</math> (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: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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 [[special linear group of degree two|degree two]] <math>SL(2,q)</math> over a finite field of size <math>q</math> || <math>q = 5</math>, i.e., [[field:F5]] || Case <math>q</math> odd: <math>q + 4</math><br>Case <math>q</math> even: <math>q + 1</math> || [[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 <math>q + 4 = 5 + 4 = 9</math> || [[#Interpretation as special linear group of degree two]] | ||

|- | |- | ||

− | | | + | | [[double cover of alternating group]] <math>2 \cdot A_n</math> || <math>n = 5</math>, i.e., the group is double cover of [[alternating group:A5]] || (number of unordered integer partitions of <math>n</math>) + 3(number of partitions of <math>n</math> into distinct odd parts) - (number of partitions of <math>n</math> 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 <matH>n = 5</math>, the three numbers to calculate are respectively 7,1,1. So, we get <math>7 + 3(1) - 1 = 9</math>. || [[#Interpretation as double cover of alternating group]] |

|- | |- | ||

− | | | + | | [[binary von Dyck group]] with parameters <math>p,q,r</math> satisfying <math>1/p + 1/q + 1/r > 1</math> || <math>(p,q,r) = (5,3,2)</math> || <math>p + q + r - 1</math> || || <math>p +q + r - 1 = 5 + 3 + 2 - 1 = 9</math> || [[#Interpretation as binary von Dyck group]] |

|} | |} |

## 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 |