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 ![]() |
![]() ![]() |
![]() ![]() |
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 ![]() ![]() |
degree ![]() ![]() |
![]() |
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 ![]() |
![]() |
![]() |
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 |
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1 | ![]() |
2 |
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4 | [SHOW MORE] | 3 |
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4 | [SHOW MORE] | 3 |
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4 | [SHOW MORE] | 6 |
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4 | [SHOW MORE] | 6 |
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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 |
---|---|---|---|---|
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1 | 1 | 1 | 1 |
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1 | 1 | 1 | 2 |
![]() ![]() |
2 | 4 | 8 | 3 |
![]() ![]() |
2 | 4 | 8 | 6 |
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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 | ![]() ![]() |
![]() ![]() |
![]() ![]() |
1 | 1 | 2 | 2 | 2 | 2 | ![]() ![]() |
Not diagonal, Jordan block of size two | ![]() ![]() |
![]() ![]() |
![]() ![]() |
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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 ![]() |
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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 ![]() ![]() |
Conclusion: does the fiber in ![]() ![]() |
Total number of conjugacy classes in ![]() |
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 ![]() ![]() |
Conclusion: does the fiber in ![]() ![]() |
Total number of conjugacy classes in ![]() |
Size of each conjugacy class | Size formula (we take the size formula in ![]() |
Total number of elements (= twice the size of the ![]() |
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) | ![]() ![]() |
2 + 2 | 2 (2 times) | Yes | Yes | No | No | 1 | 6 | ![]() |
6 | 4 | ![]() |
3 + 1 | 3 (1 time), 1 (1 time) | No | No | Yes | Yes | 4 | 4 | ![]() |
16 | 3 (2 classes) 6 (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 ![]() ![]() |
![]() ![]() |
![]() ![]() ![]() ![]() |
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 ![]() ![]() |
![]() ![]() |
(number of unordered integer partitions of ![]() ![]() ![]() |
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 ![]() |
![]() |
![]() |
See element structure of binary von Dyck groups | ![]() |
#Interpretation as binary von Dyck group |