Element structure of projective special linear group of degree two over a finite field
This article describes the element structure of projective special linear group of degree two over a finite field of order and characteristic
. Some aspects of this discussion, with suitable infinitary analogues of cardinality, carry over to infinite fields and fields of infinite characteristic.
Contents
Summary
Item | Value |
---|---|
conjugacy class sizes | Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
number of conjugacy classes | Case ![]() ![]() Case ![]() ![]() equals number of irreducible representations, see also linear representation theory of projective special linear group of degree two over a finite field |
number of ![]() ![]() |
![]() equals the number of irreducible representations in that characteristic, see also modular representation theory of projective special linear group of degree two over a finite field in its defining characteristic |
order | General formula: ![]() Case ![]() ![]() Case ![]() ![]() |
exponent | Case ![]() ![]() Case ![]() ![]() |
Particular cases
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Case on ![]() |
Group | order of the group (= ![]() ![]() ![]() ![]() |
sizes of conjugacy classes (ascending order) | number of conjugacy classes (= ![]() ![]() ![]() ![]() |
element structure page |
---|---|---|---|---|---|---|---|
2 | 2 | even | symmetric group:S3 | 6 | 1,2,3 | 3 | element structure of symmetric group:S3 |
3 | 3 | 3 mod 4 | alternating group:A4 | 12 | 1,3,4,4 | 4 | element structure of alternating group:A4 |
4 | 2 | even | alternating group:A5 | 60 | 1,12,15,20,20 | 5 | element structure of alternating group:A5 |
5 | 5 | 1 mod 4 | alternating group:A5 | 60 | 1,12,15,20,20 | 5 | element structure of alternating group:A5 |
7 | 7 | 3 mod 4 | projective special linear group:PSL(3,2) | 168 | 1,21,24,24,42,56 | 6 | element structure of projective special linear group:PSL(3,2) |
8 | 2 | even | projective special linear group:PSL(2,8) | 504 | 1,56,56,56,56,63,72,72,72 | 9 | element structure of projective special linear group:SL(2,8) |
9 | 3 | 1 mod 4 | alternating group:A6 | 360 | 1,40,40,45,72,72,90 | 7 | element structure of alternating group:A6 |
Conjugacy class structure
See also element structure of special linear group of degree two#Conjugacy class structure.
Number of conjugacy classes
As we know in general, number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size, the degree of this PORC function is one less than the degree of matrices, and we make cases based on the congruence classes modulo the degree of matrices. Thus, we expect that the number of conjugacy classes is a PORC function of the field size of degree 2 - 1 = 1, and we need to make cases based on the congruence class of the field size modulo 2. Moreover, the general theory also tells us that the polynomial function of depends only on the value of
, which in turn can be determined by the congruence class of
mod
(with
here).
Value of ![]() |
Corresponding congruence classes of ![]() |
Number of conjugacy classes (polynomial of degree 2 - 1 = 1 in ![]() |
Additional comments |
---|---|---|---|
1 | 0 mod 2 (e.g., ![]() |
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In this case, we have an isomorphism between linear groups when degree power map is bijective, so ![]() |
2 | 1 mod 2 (e.g., ![]() |
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Case where
is odd,
divides 
Nature of conjugacy class upstairs in ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Orders of elements |
---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
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1 | 1 | 1 | 1 |
Not diagonal, has Jordan block of size two | ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
2 | ![]() |
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Diagonalizable over ![]() ![]() |
![]() ![]() |
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1 | ![]() |
2 |
Diagonalizable over ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() ![]() ![]() |
Same as characteristic polynomial | ![]() |
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divisor of ![]() ![]() |
Diagonalizable over ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
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divisor of ![]() ![]() ![]() |
Total | NA | NA | NA | NA | ![]() |
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Case where
is odd,
does not divides 
Nature of conjugacy class upstairs in ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Order of elements |
---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
1 | 1 | 1 | 1 |
Diagonalizable over ![]() ![]() ![]() |
Square roots of ![]() |
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1 | ![]() |
2 |
Not diagonal, has Jordan block of size two | ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
2 | ![]() |
![]() |
Diagonalizable over ![]() ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() ![]() ![]() ![]() |
Same as characteristic polynomial | ![]() |
![]() |
![]() |
divisor of ![]() ![]() |
Diagonalizable over ![]() |
![]() ![]() ![]() ![]() ![]() |
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divisor of ![]() ![]() |
Total | NA | NA | NA | NA | ![]() |
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Case where 
In this case, the natural surjective map from the special linear group of degree two to the projective special linear group of degree two is an isomorphism, so the conjugacy class structure of both groups is the same. Details below:
Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ![]() |
Splits in ![]() ![]() |
---|---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
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1 | 1 | 1 | Yes | Yes | No |
Diagonalizable over ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() |
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![]() |
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Yes | No | No |
Not diagonal, has Jordan block of size two | ![]() |
![]() |
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1 | ![]() |
No | No | No |
Diagonalizable over ![]() |
![]() ![]() |
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Yes | Yes | No |
Total | NA | NA | NA | NA | ![]() |
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