Element structure of groups of order 243: Difference between revisions

Revision as of 18:40, 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
Sylow subgroup of holomorph of Z27	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]

List([1..67],i ->[OrderStatistics(SmallGroup(243,i)),i]);

after first defining the OrderStatistics function (follow link for GAP code for function definition). The output is:

[ [ [ 1, 2, 6, 18, 54, 162 ], 1 ], [ [ 1, 26, 216, 0, 0, 0 ], 2 ], [ [ 1, 134, 108, 0, 0, 0 ], 3 ], [ [ 1, 80, 162, 0, 0, 0 ], 4 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 5 ], [ [ 1, 80, 162, 0, 0, 0 ], 6 ], [ [ 1, 26, 216, 0, 0, 0 ], 7 ], [ [ 1, 26, 216, 0, 0, 0 ], 8 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 9 ], [ [ 1, 8, 72, 162, 0, 0 ], 10 ], [ [ 1, 8, 72, 162, 0, 0 ], 11 ], [ [ 1, 26, 54, 162, 0, 0 ], 12 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 13 ], [ [ 1, 26, 216, 0, 0, 0 ], 14 ], [ [ 1, 26, 216, 0, 0, 0 ], 15 ], [ [ 1, 26, 54, 162, 0, 0 ], 16 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 17 ], [ [ 1, 26, 216, 0, 0, 0 ], 18 ], [ [ 1, 26, 54, 162, 0, 0 ], 19 ], [ [ 1, 26, 54, 162, 0, 0 ], 20 ],
  [ [ 1, 8, 72, 162, 0, 0 ], 21 ], [ [ 1, 8, 72, 162, 0, 0 ], 22 ], [ [ 1, 8, 18, 54, 162, 0 ], 23 ], [ [ 1, 8, 18, 54, 162, 0 ], 24 ],
  [ [ 1, 62, 180, 0, 0, 0 ], 25 ], [ [ 1, 170, 72, 0, 0, 0 ], 26 ], [ [ 1, 8, 234, 0, 0, 0 ], 27 ], [ [ 1, 116, 126, 0, 0, 0 ], 28 ],
  [ [ 1, 8, 234, 0, 0, 0 ], 29 ], [ [ 1, 62, 180, 0, 0, 0 ], 30 ], [ [ 1, 26, 216, 0, 0, 0 ], 31 ], [ [ 1, 80, 162, 0, 0, 0 ], 32 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 33 ], [ [ 1, 26, 216, 0, 0, 0 ], 34 ], [ [ 1, 80, 162, 0, 0, 0 ], 35 ], [ [ 1, 26, 216, 0, 0, 0 ], 36 ],
  [ [ 1, 242, 0, 0, 0, 0 ], 37 ], [ [ 1, 80, 162, 0, 0, 0 ], 38 ], [ [ 1, 80, 162, 0, 0, 0 ], 39 ], [ [ 1, 80, 162, 0, 0, 0 ], 40 ],
  [ [ 1, 26, 216, 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, 54, 162, 0, 0 ], 48 ],
  [ [ 1, 26, 54, 162, 0, 0 ], 49 ], [ [ 1, 26, 54, 162, 0, 0 ], 50 ], [ [ 1, 134, 108, 0, 0, 0 ], 51 ], [ [ 1, 80, 162, 0, 0, 0 ], 52 ],
  [ [ 1, 188, 54, 0, 0, 0 ], 53 ], [ [ 1, 26, 216, 0, 0, 0 ], 54 ], [ [ 1, 80, 162, 0, 0, 0 ], 55 ], [ [ 1, 134, 108, 0, 0, 0 ], 56 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 57 ], [ [ 1, 188, 54, 0, 0, 0 ], 58 ], [ [ 1, 26, 216, 0, 0, 0 ], 59 ], [ [ 1, 80, 162, 0, 0, 0 ], 60 ],
  [ [ 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 ] ]

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	9	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
Direct product of Z27 and Z9	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
Sylow subgroup of holomorph of Z27	22	1	9	81	243	243	243
Direct product of Z81 and Z3	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
Direct product of Z9 and Z9 and Z3	31	1	27	243	243	243	243
	32	1	81	243	243	243	243
	33	1	27	243	243	243	243
	34	1	27	243	243	243	243
	35	1	81	243	243	243	243
	36	1	27	243	243	243	243
	37	1	243	243	243	243	243
	38	1	81	243	243	243	243
	39	1	81	243	243	243	243
	40	1	81	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	243	243	243	243
Direct product of Z27 and E9	48	1	27	81	243	243	243
	49	1	27	81	243	243	243
	50	1	27	81	243	243	243
	51	1	135	243	243	243	243
	52	1	81	243	243	243	243
	53	1	189	243	243	243	243
	54	1	27	243	243	243	243
	55	1	81	243	243	243	243
	56	1	135	243	243	243	243
	57	1	81	243	243	243	243
	58	1	189	243	243	243	243
	59	1	27	243	243	243	243
	60	1	81	243	243	243	243
Direct product of Z9 and E27	61	1	81	243	243	243	243
	62	1	243	243	243	243	243
	63	1	81	243	243	243	243
	64	1	81	243	243	243	243
	65	1	243	243	243	243	243
	66	1	81	243	243	243	243
Elementary abelian group:E243	67	1	243	243	243	243	243

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

List([1..67],i ->[OrderStatisticsCumulative(SmallGroup(243,i)),i]);

after first defining the OrderStatisticsCumulative function (follow link for GAP code for function definition). The output is:

[ [ [ 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 ] ]

@@ Line 52: / Line 52: @@
 | ||21 || 1 || 8 || 72 || 162 || 0 || 0
 |-
-| ||22 || 1 || 8 || 72 || 162 || 0 || 0
+| [[Sylow subgroup of holomorph of Z27]] ||22 || 1 || 8 || 72 || 162 || 0 || 0
 |-
 | [[Direct product of Z81 and Z3]] ||23 || 1 || 8 || 18 || 54 || 162 || 0
@@ Line 216: / Line 216: @@
 | ||21 || 1 || 9 || 81 || 243 || 243 || 243
 |-
-| ||22 || 1 || 9 || 81 || 243 || 243 || 243
+| [[Sylow subgroup of holomorph of Z27]] ||22 || 1 || 9 || 81 || 243 || 243 || 243
 |-
 | [[Direct product of Z81 and Z3]] ||23 || 1 || 9 || 27 || 81 || 243 || 243