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== | |
+ | |||
+ | {| 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 | ||
+ | |} | ||
==Conjugacy class structure== | ==Conjugacy class structure== |
Revision as of 01:12, 31 May 2012
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 |
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
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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12 | 4 | 4 | ![]() |
48 | [SHOW MORE] |
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. | ![]() ![]() ![]() |
![]() ![]() |
![]() ![]() |
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20 | ![]() |
2 | ![]() |
40 | ![]() ![]() |
Diagonalizable over field:F5 with distinct diagonal entries | ![]() |
![]() |
![]() |
![]() |
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 ![]() ![]() |
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 | 1 (5 times) | No | Yes | No | Yes | 2 | 1 | ![]() |
2 | 1 (1 class), 2 (1 class) | ![]() ![]() |
2 + 2 + 1 | 2 (2 times), 1 (1 time) | Yes | Yes | No | No | 1 | 30 | ![]() |
30 | 4 | ![]() |
3 + 1 + 1 | 3 (1 time), 1 (2 times) | No | Yes | No | Yes | 2 | 20 | ![]() |
40 | 3 (1 class) 6 (1 class) |
![]() ![]() |
5 | 5 (1 time) | No | No | Yes | Yes | 4 | 12 | ![]() |
48 | 5 (2 classes), 10 (2 classes) | ![]() ![]() |
Total | -- | -- | -- | -- | -- | 9 | -- | -- | 120 | -- | -- |