Element structure of groups of order 243: Difference between revisions

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===Order statistics raw data===
===Order statistics raw data===
Here are the order statistics (non-cumulative version):


{| class="sortable" border="1"
{| class="sortable" border="1"
Line 166: Line 168:
   [ [ 1, 80, 162, 0, 0, 0 ], 61 ], [ [ 1, 242, 0, 0, 0, 0 ], 62 ], [ [ 1, 80, 162, 0, 0, 0 ], 63 ], [ [ 1, 80, 162, 0, 0, 0 ], 64 ],
   [ [ 1, 80, 162, 0, 0, 0 ], 61 ], [ [ 1, 242, 0, 0, 0, 0 ], 62 ], [ [ 1, 80, 162, 0, 0, 0 ], 63 ], [ [ 1, 80, 162, 0, 0, 0 ], 64 ],
   [ [ 1, 242, 0, 0, 0, 0 ], 65 ], [ [ 1, 80, 162, 0, 0, 0 ], 66 ], [ [ 1, 242, 0, 0, 0, 0 ], 67 ] ]</pre></toggledisplay>
   [ [ 1, 242, 0, 0, 0, 0 ], 65 ], [ [ 1, 80, 162, 0, 0, 0 ], 66 ], [ [ 1, 242, 0, 0, 0, 0 ], 67 ] ]</pre></toggledisplay>
Here are the order statistics (cumulative version):
{| class="sortable" border="1"
! Group !! Second part of GAP ID !! 1st roots !! 3rd roots !! 9th roots !! 27th roots !! 81st roots !! 243th roots
|-
| || 1 || 1 || 3 || 8 || 27 || 81 || 243
|-
| || 2 || 1 || 27 || 243 || 243 || 243 || 243
|-
| || 3 || 1 || 135 || 243 || 243 || 243 || 243
|-
| || 4 || 1 || 81 || 243 || 243 || 243 || 243
|-
| || 5 || 1 || 27 || 243 || 243 || 243 || 243
|-
| || 6 || 1 || 81 || 243 || 243 || 243 || 243
|-
| || 7 || 1 || 27 || 243 || 243 || 243 || 243
|-
| || 8 || 1 || 27 || 243 || 243 || 243 || 243
|-
| || 9 || 1 || 27 || 243 || 243 || 243 || 243
|-
| ||10 || 1 || 9 || 81 || 243 || 243 || 243
|-
| ||11 || 1 || 9 || 81 || 243 || 243 || 243
|-
| ||12 || 1 || 27 || 81 || 243 || 243 || 243
|-
| ||13 || 1 || 81 || 243|| 243 || 243 || 243
|-
| ||14 || 1 || 27 || 243|| 243 || 243 || 243
|-
| ||15 || 1 || 27 || 243|| 243 || 243 || 243
|-
| ||16 || 1 || 27 || 81 || 243 || 243 || 243
|-
| ||17 || 1 || 81 || 243 || 243 || 243 || 243
|-
| ||18 || 1 || 27 || 243 || 243 || 243 || 243
|-
| ||19 || 1 || 27 || 81 || 243 || 243 || 243
|-
| ||20 || 1 || 27 || 81 || 243 || 243 || 243
|-
| ||21 || 1 || 9 || 81 || 243 || 243 || 243
|-
| ||22 || 1 || 9 || 81 || 243 || 243 || 243
|-
| ||23 || 1 || 9 || 27 || 81 || 243 || 243
|-
| ||24 || 1 || 9 || 27 || 81 || 243 || 243
|-
| ||25 || 1 || 63 || 243||243|| 243 || 243
|-
| ||26 || 1 || 171||243 ||243|| 243 || 243
|-
| ||27 || 1 || 9 || 243 || 243 || 243||243
|-
| ||28 || 1 ||117|| 243 || 243 || 243||243
|-
| ||29 || 1 || 9 || 243 || 243 || 243||243
|-
| ||30 || 1 || 63 || 243|| 243 || 243||243
|}
|}
Here is the GAP code to generate these order statistics:<toggledisplay>
<tt>List([1..67],i ->[OrderStatisticsCumulative(SmallGroup(243,i)),i]);</tt>
after first defining the [[GAP:OrderStatisticsCumulative|OrderStatisticsCumulative]] function (follow link for GAP code for function definition). The output is:
<pre>[ [ [ 1, 3, 9, 27, 81, 243 ], 1 ], [ [ 1, 27, 243, 243, 243, 243 ], 2 ], [ [ 1, 135, 243, 243, 243, 243 ], 3 ], [ [ 1, 81, 243, 243, 243, 243 ], 4 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 5 ], [ [ 1, 81, 243, 243, 243, 243 ], 6 ], [ [ 1, 27, 243, 243, 243, 243 ], 7 ], [ [ 1, 27, 243, 243, 243, 243 ], 8 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 9 ], [ [ 1, 9, 81, 243, 243, 243 ], 10 ], [ [ 1, 9, 81, 243, 243, 243 ], 11 ], [ [ 1, 27, 81, 243, 243, 243 ], 12 ],
  [ [ 1, 81, 243, 243, 243, 243 ], 13 ], [ [ 1, 27, 243, 243, 243, 243 ], 14 ], [ [ 1, 27, 243, 243, 243, 243 ], 15 ],
  [ [ 1, 27, 81, 243, 243, 243 ], 16 ], [ [ 1, 81, 243, 243, 243, 243 ], 17 ], [ [ 1, 27, 243, 243, 243, 243 ], 18 ], [ [ 1, 27, 81, 243, 243, 243 ], 19 ]
    , [ [ 1, 27, 81, 243, 243, 243 ], 20 ], [ [ 1, 9, 81, 243, 243, 243 ], 21 ], [ [ 1, 9, 81, 243, 243, 243 ], 22 ], [ [ 1, 9, 27, 81, 243, 243 ], 23 ],
  [ [ 1, 9, 27, 81, 243, 243 ], 24 ], [ [ 1, 63, 243, 243, 243, 243 ], 25 ], [ [ 1, 171, 243, 243, 243, 243 ], 26 ], [ [ 1, 9, 243, 243, 243, 243 ], 27 ],
  [ [ 1, 117, 243, 243, 243, 243 ], 28 ], [ [ 1, 9, 243, 243, 243, 243 ], 29 ], [ [ 1, 63, 243, 243, 243, 243 ], 30 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 31 ], [ [ 1, 81, 243, 243, 243, 243 ], 32 ], [ [ 1, 27, 243, 243, 243, 243 ], 33 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 34 ], [ [ 1, 81, 243, 243, 243, 243 ], 35 ], [ [ 1, 27, 243, 243, 243, 243 ], 36 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 37 ], [ [ 1, 81, 243, 243, 243, 243 ], 38 ], [ [ 1, 81, 243, 243, 243, 243 ], 39 ],
  [ [ 1, 81, 243, 243, 243, 243 ], 40 ], [ [ 1, 27, 243, 243, 243, 243 ], 41 ], [ [ 1, 27, 243, 243, 243, 243 ], 42 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 43 ], [ [ 1, 27, 243, 243, 243, 243 ], 44 ], [ [ 1, 27, 243, 243, 243, 243 ], 45 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 46 ], [ [ 1, 27, 243, 243, 243, 243 ], 47 ], [ [ 1, 27, 81, 243, 243, 243 ], 48 ], [ [ 1, 27, 81, 243, 243, 243 ], 49 ]
    , [ [ 1, 27, 81, 243, 243, 243 ], 50 ], [ [ 1, 135, 243, 243, 243, 243 ], 51 ], [ [ 1, 81, 243, 243, 243, 243 ], 52 ],
  [ [ 1, 189, 243, 243, 243, 243 ], 53 ], [ [ 1, 27, 243, 243, 243, 243 ], 54 ], [ [ 1, 81, 243, 243, 243, 243 ], 55 ],
  [ [ 1, 135, 243, 243, 243, 243 ], 56 ], [ [ 1, 81, 243, 243, 243, 243 ], 57 ], [ [ 1, 189, 243, 243, 243, 243 ], 58 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 59 ], [ [ 1, 81, 243, 243, 243, 243 ], 60 ], [ [ 1, 81, 243, 243, 243, 243 ], 61 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 62 ], [ [ 1, 81, 243, 243, 243, 243 ], 63 ], [ [ 1, 81, 243, 243, 243, 243 ], 64 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 65 ], [ [ 1, 81, 243, 243, 243, 243 ], 66 ], [ [ 1, 243, 243, 243, 243, 243 ], 67 ] ]</pre></toggledisplay>

Revision as of 18:04, 2 July 2010

Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Order statistics raw data

Here are the order statistics (non-cumulative version):

Group Second part of GAP ID Order 1 Order 3 Order 9 Order 27 Order 81 Order 243
Cyclic group:Z243 1 1 2 6 18 54 162
2 1 26 216 0 0 0
3 1 134 108 0 0 0
4 1 80 162 0 0 0
5 1 26 216 0 0 0
6 1 80 162 0 0 0
7 1 26 216 0 0 0
8 1 26 216 0 0 0
9 1 26 216 0 0 0
Direct product of Z27 and Z9 10 1 8 72 162 0 0
11 1 8 72 162 0 0
12 1 26 54 162 0 0
13 1 80 162 0 0 0
14 1 26 216 0 0 0
15 1 26 216 0 0 0
16 1 26 54 162 0 0
17 1 80 162 0 0 0
18 1 26 216 0 0 0
19 1 26 54 162 0 0
20 1 26 54 162 0 0
21 1 8 72 162 0 0
22 1 8 72 162 0 0
Direct product of Z81 and Z3 23 1 8 18 54 162 0
24 1 8 18 54 162 0
25 1 62 180 0 0 0
26 1 170 72 0 0 0
27 1 8 234 0 0 0
28 1 116 126 0 0 0
29 1 8 234 0 0 0
30 1 62 180 0 0 0
Direct product of Z9 and Z9 and Z3 31 1 26 216 0 0 0
32 1 80 162 0 0 0
33 1 26 216 0 0 0
34 1 26 216 0 0 0
35 1 80 162 0 0 0
36 1 26 216 0 0 0
37 1 242 0 0 0 0
38 1 80 162 0 0 0
39 1 80 162 0 0 0
40 1 80 162 0 0 0
41 1 26 216 0 0 0
42 1 26 216 0 0 0
43 1 26 216 0 0 0
44 1 26 216 0 0 0
45 1 26 216 0 0 0
46 1 26 216 0 0 0
47 1 26 216 0 0 0
Direct product of Z27 and E9 48 1 26 54 162 0 0
49 1 26 54 162 0 0
50 1 26 54 162 0 0
51 1 134 108 0 0 0
52 1 80 162 0 0 0
53 1 188 54 0 0 0
54 1 26 216 0 0 0
55 1 80 162 0 0 0
56 1 134 108 0 0 0
57 1 80 162 0 0 0
58 1 188 54 0 0 0
59 1 26 216 0 0 0
60 1 80 162 0 0 0
Direct product of Z9 and E27 61 1 80 162 0 0 0
62 1 242 0 0 0 0
63 1 80 162 0 0 0
64 1 80 162 0 0 0
65 1 242 0 0 0 0
66 1 80 162 0 0 0
Elementary abelian group:E243 67 1 242 0 0 0 0

Here is the GAP code to generate these order statistics:[SHOW MORE]

Here are the order statistics (cumulative version):

Group Second part of GAP ID 1st roots 3rd roots 9th roots 27th roots 81st roots 243th roots
1 1 3 8 27 81 243
2 1 27 243 243 243 243
3 1 135 243 243 243 243
4 1 81 243 243 243 243
5 1 27 243 243 243 243
6 1 81 243 243 243 243
7 1 27 243 243 243 243
8 1 27 243 243 243 243
9 1 27 243 243 243 243
10 1 9 81 243 243 243
11 1 9 81 243 243 243
12 1 27 81 243 243 243
13 1 81 243 243 243 243
14 1 27 243 243 243 243
15 1 27 243 243 243 243
16 1 27 81 243 243 243
17 1 81 243 243 243 243
18 1 27 243 243 243 243
19 1 27 81 243 243 243
20 1 27 81 243 243 243
21 1 9 81 243 243 243
22 1 9 81 243 243 243
23 1 9 27 81 243 243
24 1 9 27 81 243 243
25 1 63 243 243 243 243
26 1 171 243 243 243 243
27 1 9 243 243 243 243
28 1 117 243 243 243 243
29 1 9 243 243 243 243
30 1 63 243 243 243 243

|}

Here is the GAP code to generate these order statistics:[SHOW MORE]

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