Classification of finite simple groups - Revision history

R-a-jones: /* The twenty-six sporadic simple groups */ put the table into a template so it can be reused elsewhere easily

2023-11-21T11:44:28Z

The twenty-six sporadic simple groups: put the table into a template so it can be reused elsewhere easily

@@ Line 58: / Line 58: @@
 ==The twenty-six sporadic simple groups==
-{| class="sortable" border="1"
+{{Sporadic simple groups table}}
 ==List of simple non-abelian groups of small order==

Vipul: /* Collisions between families */

2012-09-12T23:44:29Z

Collisions between families

← Older revision		Revision as of 23:44, 12 September 2012
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	\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.		\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.
	\|-		\|-
	\| <math>B_2(q) \cong C_2(q)</math> for all <math>q</math>. In other words, <math>\Omega_5(q) \cong PSp(4,q)</~~mth~~> for all <math>q</math>. \|\| We denote the group as <math>B_2(q)</math>.		\| <math>B_2(q) \cong C_2(q)</math> for all <math>q</math>. In other words, <math>\Omega_5(q) \cong PSp(4,q)</math> for all <math>q</math>. \|\| We denote the group as <math>B_2(q)</math>.
	\|-		\|-
	\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| We denote the group as <math>B_n(2^m)</math>.		\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| We denote the group as <math>B_n(2^m)</math>.

Vipul: /* Collisions between families */

2012-08-30T19:15:50Z

Collisions between families

← Older revision		Revision as of 19:15, 30 August 2012
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	\|-		\|-
	\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.		\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.
			\|-
			\| <math>B_2(q) \cong C_2(q)</math> for all <math>q</math>. In other words, <math>\Omega_5(q) \cong PSp(4,q)</mth> for all <math>q</math>. \|\| We denote the group as <math>B_2(q)</math>.
	\|-		\|-
	\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| We denote the group as <math>B_n(2^m)</math>.		\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| We denote the group as <math>B_n(2^m)</math>.

Vipul: /* Collisions between families */

2012-08-30T19:11:24Z

Collisions between families

← Older revision		Revision as of 19:11, 30 August 2012
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	\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.		\| <math>A_1(q) \cong B_1(q) \cong C_1(q)</math>. In other words, <math>PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q)</math> for all <math>q</math>. \|\| We denote the group as <math>A_1(q)</math> or <math>PSL(2,q)</math>.
	\|-		\|-
	\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| ~~Unclear, either is fine~~.		\| <math>B_n(2^m) \cong C_n(2^m)</math> for all <math>m,n</math>. In other words, <math>\Omega_{2n+1}(2^m) \cong PSp(2n,2^m)</math> for all <math>m</math>. Note that the <math>n = 1</math> case is already covered in the preceding collision. \|\| We denote the group as <math>B_n(2^m)</math>.
	\|}		\|}

Vipul: /* The twenty-six sporadic simple groups */

2012-05-21T21:40:14Z

The twenty-six sporadic simple groups

← Older revision		Revision as of 21:40, 21 May 2012
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	\| [[monster group]] \|\| <math>M</math> \|\| 808017424794512875886459904961710757005754368000000000 \|\| <toggledisplay><math>2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71</math></toggledisplay> \|\| 194		\| [[monster group]] \|\| <math>M</math> \|\| 808017424794512875886459904961710757005754368000000000 \|\| <toggledisplay><math>2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71</math></toggledisplay> \|\| 194
	\|}		\|}

	~~More to be added~~

	==List of simple non-abelian groups of small order==		==List of simple non-abelian groups of small order==

Vipul: /* List of simple non-abelian groups of small order */

2012-05-21T02:54:21Z

List of simple non-abelian groups of small order

← Older revision		Revision as of 02:54, 21 May 2012
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	==List of simple non-abelian groups of small order==		==List of simple non-abelian groups of small order==

			{{further\|[[List of simple non-abelian groups of small order]]}}

	The simple ''abelian'' groups are precisely the groups of prime order, and there is one such group for each prime number.		The simple ''abelian'' groups are precisely the groups of prime order, and there is one such group for each prime number.

Vipul: /* The twenty-six sporadic simple groups */

2012-05-21T02:21:10Z

The twenty-six sporadic simple groups

← Older revision		Revision as of 02:21, 21 May 2012
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	\| [[Conway group:Co3]] \|\| <math>\operatorname{Co}_3</math> \|\| 495766656000 \|\| <math>2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23</math> \|\| 42		\| [[Conway group:Co3]] \|\| <math>\operatorname{Co}_3</math> \|\| 495766656000 \|\| <math>2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23</math> \|\| 42
	\|-		\|-
	\| [[Fischer group:Fi22]] \|\| <math>~~Fi_~~{22}</math> \|\| 64561751654400 \|\| <math>2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13</math> \|\| 65		\| [[Fischer group:Fi22]] \|\| <math>\operatorname{Fi}_{22}</math> \|\| 64561751654400 \|\| <math>2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13</math> \|\| 65
	\|-		\|-
	\| [[Fischer group:Fi23]] \|\| <math>~~Fi_~~{23}</math> \|\| 4089470473293004800 \|\| <math>2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23</math> \|\| 98		\| [[Fischer group:Fi23]] \|\| <math>\operatorname{Fi}_{23}</math> \|\| 4089470473293004800 \|\| <math>2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23</math> \|\| 98
	\|-		\|-
	\| [[derived subgroup of Fischer group:Fi24]] \|\| <math>F_{24}</math> \|\| 1255205709190661721292800 \|\| <math>2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29</math> \|\| 183		\| [[derived subgroup of Fischer group:Fi24]] \|\| <math>F_{24}</math> \|\| 1255205709190661721292800 \|\| <math>2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29</math> \|\| 183

Vipul: /* The twenty-six sporadic simple groups */

2012-05-21T02:17:13Z

The twenty-six sporadic simple groups

← Older revision		Revision as of 02:17, 21 May 2012
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	\|-		\|-
	\| [[Conway group:Co3]] \|\| <math>\operatorname{Co}_3</math> \|\| 495766656000 \|\| <math>2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23</math> \|\| 42		\| [[Conway group:Co3]] \|\| <math>\operatorname{Co}_3</math> \|\| 495766656000 \|\| <math>2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23</math> \|\| 42
			\|-
			\| [[Fischer group:Fi22]] \|\| <math>Fi_{22}</math> \|\| 64561751654400 \|\| <math>2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13</math> \|\| 65
			\|-
			\| [[Fischer group:Fi23]] \|\| <math>Fi_{23}</math> \|\| 4089470473293004800 \|\| <math>2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23</math> \|\| 98
			\|-
			\| [[derived subgroup of Fischer group:Fi24]] \|\| <math>F_{24}</math> \|\| 1255205709190661721292800 \|\| <math>2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29</math> \|\| 183
	\|-		\|-
	\| [[Higman-Sims group]] \|\| <math>HS</math> \|\| 44352000 \|\| <math>2^9 \cdot 3^2 \cdot 5^3 \cdot 7 \cdot 11</math> \|\| 24		\| [[Higman-Sims group]] \|\| <math>HS</math> \|\| 44352000 \|\| <math>2^9 \cdot 3^2 \cdot 5^3 \cdot 7 \cdot 11</math> \|\| 24

Vipul: /* The twenty-six sporadic simple groups */

2012-05-21T01:24:11Z

The twenty-six sporadic simple groups

← Older revision		Revision as of 01:24, 21 May 2012
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	\|-		\|-
	\| [[Rudvalis group]] \|\| <math>Ru</math> \|\| 145926144000 \|\|<math>2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29</math> \|\| 36		\| [[Rudvalis group]] \|\| <math>Ru</math> \|\| 145926144000 \|\|<math>2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29</math> \|\| 36
			\|-
			\| [[Suzuki sporadic group]] \|\| <math>Suz</math> \|\| 448345497600 \|\| <math>2^{13} \cdot 3^7 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13</math> \|\| 43
			\|-
			\| [[O'Nan group]] \|\| <math>ON</math> \|\| 460815505920 \|\| <math>2^9 \cdot 3^4 \cdot 5 \cdot 7^3 \cdot 11 \cdot 19 \cdot 31</math> \|\| 30
	\|-		\|-
	\| [[Harada-Norton group]] \|\| <math>HN</math> \|\| 273030912000000 \|\| <math>2^{14} \cdot 3^6 \cdot 5^6 \cdot 7 \cdot 11 \cdot 19</math> \|\| 54		\| [[Harada-Norton group]] \|\| <math>HN</math> \|\| 273030912000000 \|\| <math>2^{14} \cdot 3^6 \cdot 5^6 \cdot 7 \cdot 11 \cdot 19</math> \|\| 54

Vipul: /* The twenty-six sporadic simple groups */

2012-05-21T01:20:04Z

The twenty-six sporadic simple groups

← Older revision		Revision as of 01:20, 21 May 2012
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	\| [[McLaughlin group]] \|\| <math>McL</math> \|\| 898128000 \|\| <math>2^7 \cdot 3^6 \cdot 5 \cdot 7 \cdot 11</math> \|\| 24		\| [[McLaughlin group]] \|\| <math>McL</math> \|\| 898128000 \|\| <math>2^7 \cdot 3^6 \cdot 5 \cdot 7 \cdot 11</math> \|\| 24
	\|-		\|-
	\| [[Held group]] \|\| <math>He</math> \|\| 4030387200 \|\| ~~<toggledisplay>~~<math>2^{10} \cdot 3^3 \cdot 5^2 \cdot 7^3 \cdot 17</math~~></toggledisplay~~> \|\| 33		\| [[Held group]] \|\| <math>He</math> \|\| 4030387200 \|\| <math>2^{10} \cdot 3^3 \cdot 5^2 \cdot 7^3 \cdot 17</math> \|\| 33
	\|-		\|-
	\| [[Rudvalis group]] \|\| <math>Ru</math> \|\| 145926144000 \|\| ~~<toggledisplay>~~<math>2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29</math~~></toggledisplay~~> \|\| 36		\| [[Rudvalis group]] \|\| <math>Ru</math> \|\| 145926144000 \|\|<math>2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29</math> \|\| 36
	\|-		\|-
	\| [[Harada-Norton group]] \|\| <math>HN</math> \|\| 273030912000000 \|\| ~~<toggledisplay>~~<math>2^{14} \cdot 3^6 \cdot 5^6 \cdot 7 \cdot 11 \cdot 19</math~~></toggledisplay~~> \|\| 54		\| [[Harada-Norton group]] \|\| <math>HN</math> \|\| 273030912000000 \|\| <math>2^{14} \cdot 3^6 \cdot 5^6 \cdot 7 \cdot 11 \cdot 19</math> \|\| 54
	\|-		\|-
	\| [[Lyons group]] \|\| <math>Ly</math> \|\| 51765179004000000 \|\| ~~<toggledisplay>~~<math>2^8 \cdot 3^7 \cdot 5^6 \cdot 7 \cdot 11 \cdot 31 \cdot 37 \cdot 67</math~~></toggledisplay~~> \|\| 53		\| [[Lyons group]] \|\| <math>Ly</math> \|\| 51765179004000000 \|\| <math>2^8 \cdot 3^7 \cdot 5^6 \cdot 7 \cdot 11 \cdot 31 \cdot 37 \cdot 67</math> \|\| 53
	\|-		\|-
	\| [[Thompson group]] \|\| <math>Th</math> \|\| 90745943887872000 \|\| ~~<toggledisplay>~~<math>2^{15} \cdot 3^{10} \cdot 5^3 \cdot 7^2 \cdot 13 \cdot 19 \cdot 31</math~~></toggledisplay~~> \|\| 48		\| [[Thompson group]] \|\| <math>Th</math> \|\| 90745943887872000 \|\| <math>2^{15} \cdot 3^{10} \cdot 5^3 \cdot 7^2 \cdot 13 \cdot 19 \cdot 31</math> \|\| 48
	\|-		\|-
	\| [[baby monster group]] \|\| <math>B</math> \|\| 4154781481226426191177580544000000 \|\| <toggledisplay><math>2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13\cdot 17\cdot 19 \cdot 23 \cdot 31 \cdot 47</math></toggledisplay> \|\| 184		\| [[baby monster group]] \|\| <math>B</math> \|\| 4154781481226426191177580544000000 \|\| <toggledisplay><math>2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13\cdot 17\cdot 19 \cdot 23 \cdot 31 \cdot 47</math></toggledisplay> \|\| 184