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

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## 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 $SL(2,9)$ has order 720. with prime factorization $720 = 2^4 \cdot 3^2 \cdot 5^1 = 16 \cdot 9 \cdot 5$. 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 $q$ $q = 9$, i.e., field:F9, so the group is $SL(2,9)$ $q^3 - q$, in factored form $q(q - 1)(q + 1)$ See order formulas for linear groups of degree two, order formulas for linear groups, and special linear group of degree two $9^3 - 9 = 720$
Factored version: $9(9 - 1)(9 + 1) = 9(8)(10) = 720$
#Interpretation as special linear group of degree two
double cover of alternating group $2 \cdot A_n$ of degree $n$ degree $n = 6$, so the group is $2 \cdot A_6$ $n!$ See double cover of alternating group, element structure of double cover of alternating group $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$ #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 $q$) Size of conjugacy class ( $q = 9$) Number of such conjugacy classes (generic odd $q$) Number of such conjugacy classes ( $q = 9$) Total number of elements (generic odd $q$) Total number of elements ( $q = 9$) Representative matrices (one per conjugacy class)
Scalar $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x - 1$ or $x + 1$ 1 1 2 2 2 2 $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}$
Not diagonal, Jordan block of size two $\{ 1, 1 \}$ or $\{ -1,-1\}$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $x^2 - 2x + 1$ or $x^2 + 2x + 1$ $(q^2 - 1)/2$ 40 4 4 $2(q^2 - 1)$ 160 [SHOW MORE] where $\alpha$ 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] $q(q - 1)$ 72 $(q - 1)/2$ 4 $q(q - 1)^2/2$ 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] $q(q+1)$ 90 $(q - 3)/2$ 3 $q(q+1)(q-3)/2$ 270 --
Total NA NA NA NA NA $q + 4$ 13 $q^3 - q$ 720 NA

### Interpretation as double cover of alternating group

Further information: element structure of double cover of alternating group $SL(2,9)$ is isomorphic to $2 \cdot A_n,n = 6$. 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 $S_n$ to $A_n$ in 2? Conclusion: does the fiber in $2 \cdot A_n$ over a conjugacy class in $A_n$ split in 2? Total number of conjugacy classes in $2 \cdot A_n$ 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 $S_n$ to $A_n$ in 2? Conclusion: does the fiber in $2 \cdot A_n$ over a conjugacy class in $A_n$ split in 2? Total number of conjugacy classes in $2 \cdot A_n$ 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 $S_n$, multiply by 2, and divide by the number (1,2, or 4) two columns preceding Total number of elements (= twice the size of the $S_n$-conjugacy class) Element orders Formula for element orders
1 + 1 + 1 + 1 + 1 + 1 1 (6 times) No Yes No Yes 2 1 $\frac{2}{2} \frac{6!}{(1)^66!}$ 2 1 (1 class), 2 (1 class) $\operatorname{lcm} \{ 1 \}$ (1 class) $2 \operatorname{lcm} \{ 1 \}$ (1 class)
3 + 1 + 1 + 1 3 (1 time), 1 (3 times) No Yes No Yes 2 40 $\frac{2}{2} \frac{6!}{(3)(1)^3(3!)}$ 80 3 (1 class), 6 (1 class) $\operatorname{lcm} \{ 3,1 \}$ (1 class) $2 \operatorname{lcm} \{ 3,1 \}$ (1 class)
2 + 2 + 1 + 1 2 (2 times), 1 (2 times) Yes Yes No No 1 90 $\frac{2}{1} \frac{6!}{(2)^2(2!)(1)^2(2!)}$ 90 4 (1 class) $2 \operatorname{lcm} \{ 2, 1 \}$
5 + 1 5 (1 time), 1 (1 time) No No Yes Yes 4 72 $\frac{2}{4} \frac{6!}{(5)(1)}$ 288 5 (2 classes), 10 (2 classes) $\operatorname{lcm} \{ 5,1 \}$ (1 class) $2 \operatorname{lcm} \{ 5,1 \}$ (1 class)
4 + 2 4 (1 time), 2 (1 time) Yes No No Yes 2 90 $\frac{2}{2} \frac{6!}{(4)(2)}$ 180 8 (2 classes) $2 \operatorname{lcm} \{ 4,2 \}$ (2 classes)
3 + 3 3 (2 times) No Yes No Yes 2 40 $\frac{2}{2} \frac{6!}{(3)^22!}$ 80 3 (1 class), 6 (1 class) $\operatorname{lcm} \{ 3 \}$ (1 class) $2 \operatorname{lcm} \{ 3 \}$ (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 $SL(2,q)$ over a finite field of size $q$ $q = 9$, i.e., field:F9 Case $q$ odd: $q + 4$
Case $q$ even: $q + 1$
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 $q + 4 = 9 + 4 = 9$ #Interpretation as special linear group of degree two
double cover of alternating group $2 \cdot A_n$ $n = 6$, i.e., the group is double cover of alternating group:A6 (number of unordered integer partitions of $n$) + 3(number of partitions of $n$ into distinct odd parts) - (number of partitions of $n$ 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 $n = 6$, the three numbers to calculate are respectively 11,1,1. So, we get $11 + 3(1) - 1 = 13$. #Interpretation as double cover of alternating group