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Element structure of special linear group:SL(2,5)

3,244 bytes added, 23:47, 30 May 2012
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===Relationship with conjugacy class structure for an arbitrary Interpretation as special linear group of degree two===
{{further|[[element structure of special linear group of degree two over a finite field]]}}
| Diagonalizable over [[field:F5]] with ''distinct'' diagonal entries || <math>\{ 2,3 \}</math> || <math>x^2 + 1</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>
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| ! 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>2 \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 || -- || --
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