Symmetric group:S6 - Revision history

R-a-jones at 21:23, 15 December 2023

2023-12-15T21:23:22Z

← Older revision		Revision as of 21:23, 15 December 2023
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	{{particular group}}		{{particular group}}
			[[Category:Symmetric groups]]
	[[importance rank::2\| ]]		[[importance rank::2\| ]]
	==Definition==		==Definition==

Vipul: /* Group properties */

2013-07-07T18:04:46Z

Group properties

← Older revision		Revision as of 18:04, 7 July 2013
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	\|-		\|-
	\| [[satisfies property::rational group]] \|\| Yes \|\| See [[symmetric groups are rational]]		\| [[satisfies property::rational group]] \|\| Yes \|\| See [[symmetric groups are rational]]
			\|-
			\| [[satisfies property::N-group]] \|\| Yes \|\| See [[classification of symmetric groups that are N-groups]]
	\|}		\|}

	==Elements==		==Elements==

Vipul: /* Basic arithmetic functions */

2012-08-30T22:09:37Z

Basic arithmetic functions

← Older revision		Revision as of 22:09, 30 August 2012
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	! Function !! Value !! Similar groups !! Explanation for function value		! Function !! Value !! Similar groups !! Explanation for function value
	\|-		\|-
	\| {{arithmetic function value order\|720}} \|\| As <math>\!S_n, n = 6:</math> <math>n! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720</math><br>As <math>\! Sp(2m,q)</math>, <math>\! m = 2, q = 2</math>: <math>q^{m^2}\prod_{i=1}^m (q^{2i} - 1)</math> which becomes <math>2^4(2^2 - 1)(2^4 - 1) = 16 \cdot 3 \cdot 15 = 720</math>		\| {{arithmetic function value order\|720}} \|\| As <math>\!S_n, n = 6:</math> <math>n! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720</math><br>As <math>\! Sp(2m,q)</math>, <math>\! m = 2, q = 2</math> (see [[order formulas for symplectic groups]]): <math>q^{m^2}\prod_{i=1}^m (q^{2i} - 1)</math> which becomes <math>2^4(2^2 - 1)(2^4 - 1) = 16 \cdot 3 \cdot 15 = 720</math>
	\|-		\|-
	\| {{arithmetic function value exponent given order\|60\|720}} \|\| As <math>\! S_n, n = 6:</math> <math>\operatorname{lcm} \{ 1,2,3,4,5,6 \} = 60</math>		\| {{arithmetic function value exponent given order\|60\|720}} \|\| As <math>\! S_n, n = 6:</math> <math>\operatorname{lcm} \{ 1,2,3,4,5,6 \} = 60</math>

Vipul: /* Definition */

2012-08-30T18:34:33Z

Definition

← Older revision		Revision as of 18:34, 30 August 2012
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	* It is the [[member of family::symmetric group]] on a set of size six. In particular, it is a [[member of family::symmetric group on finite set]].		* It is the [[member of family::symmetric group]] on a set of size six. In particular, it is a [[member of family::symmetric group on finite set]].
	* It is the [[member of family::symplectic group]] <math>Sp(4,2)</math>, and hence also the [[member of family::projective symplectic group]] <math>PSp(4,2)</math>.		* It is the [[member of family::symplectic group]] <math>Sp(4,2)</math>, and hence also the [[member of family::projective symplectic group]] <math>PSp(4,2)</math> (see [[isomorphism between symplectic and projective symplectic group in characteristic two]]).

	==Arithmetic functions==		==Arithmetic functions==

Vipul: /* GAP implementation */

2012-08-19T00:47:48Z

GAP implementation

← Older revision		Revision as of 00:47, 19 August 2012
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	{{GAP ID\|720\|763}}		{{GAP ID\|720\|763}}

			===Other descriptions===

			{\| class="sortable" border="1"
			! Description !! Functions used
			\|-
			\| <tt>SymmetricGroup(6)</tt> \|\| [[GAP:SymmetricGroup\|SymmetricGroup]]
			\|-
			\| <tt>Sp(4,2)</tt> \|\| [[GAP:Sp\|Sp]]
			\|-
			\| <tt>PSp(4,2)</tt> \|\| [[GAP:PSp\|PSp]]
			\|}

Vipul: /* GAP implementation */

2012-04-30T00:44:13Z

GAP implementation

@@ Line 95: / Line 95: @@
 {{#lst:element structure of symmetric group:S6|automorphism class structure}}
 ==GAP implementation==
 {{GAP ID|720|763}}

Vipul: /* Arithmetic functions */

2012-03-23T16:58:19Z

Arithmetic functions

← Older revision		Revision as of 16:58, 23 March 2012
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	==Arithmetic functions==		==Arithmetic functions==

			{{compare and contrast arithmetic functions\|order = 720}}

			===Basic arithmetic functions===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Function !! Value !! Similar groups !! Explanation for function value		! Function !! Value !! Similar groups !! Explanation for function value
	\|-		\|-
	\| {{arithmetic function value order\|720}} \|\| As <math>\!~~S_k~~, k = 6:</math> <math>k! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720</math><br>As <math>\! Sp(2m,q)</math>, <math>\! m = 2, q = 2</math>: <math>q^{m^2}\prod_{i=1}^m (q^{2i} - 1)</math> which becomes <math>2^4(2^2 - 1)(2^4 - 1) = 16 \cdot 3 \cdot 15 = 720</math>		\| {{arithmetic function value order\|720}} \|\| As <math>\!S_n, n = 6:</math> <math>n! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720</math><br>As <math>\! Sp(2m,q)</math>, <math>\! m = 2, q = 2</math>: <math>q^{m^2}\prod_{i=1}^m (q^{2i} - 1)</math> which becomes <math>2^4(2^2 - 1)(2^4 - 1) = 16 \cdot 3 \cdot 15 = 720</math>
	\|-		\|-
	\| {{arithmetic function value exponent given order\|60\|720}} \|\| As <math>\! ~~S_k~~, k = 6:</math> <math>\operatorname{lcm} \{ 1,2,3,4,5,6 \} = 60</math>		\| {{arithmetic function value exponent given order\|60\|720}} \|\| As <math>\! S_n, n = 6:</math> <math>\operatorname{lcm} \{ 1,2,3,4,5,6 \} = 60</math>
	\|-		\|-
	\| [[derived length]] \|\| -- \|\| \|\| not a [[solvable group]]		\| [[derived length]] \|\| -- \|\| \|\| not a [[solvable group]]
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	\|-		\|-
	\| {{arithmetic function value given order\|chief length\|2\|720}} \|\| The subgroup [[alternating group:A6]] is simple and normal (see [[alternating groups are simple]]) and the quotient is simple ([[cyclic group:Z2\|cyclic of order two]])		\| {{arithmetic function value given order\|chief length\|2\|720}} \|\| The subgroup [[alternating group:A6]] is simple and normal (see [[alternating groups are simple]]) and the quotient is simple ([[cyclic group:Z2\|cyclic of order two]])
			\|}

			===Arithmetic functions of a counting nature===

			{\| class="sortable" border="1"
			! Function !! Value !! Similar groups !! Explanation for function value
	\|-		\|-
	~~\| {{arithmetic function value given order\|number of subgroups\|1455\|720}} \|\|~~		\| {{arithmetic function value given order\|number of conjugacy classes\|11\|720}} \|\| As <math>\! S_n, n = 6:</math> the number of conjugacy classes is <math>p(n) = p(6) = 11</math>, where <math>p</math> is the [[number of unordered integer partitions]]; see [[cycle type determines conjugacy class]].<br>As <math>Sp(4,q), q = 2</math> (even): <math>q^2 + 2q + 3 = 2^2 + 2(2) + 3 = 11</math>. More [[element structure of symplectic group of degree four over a finite field\|here]]<br>See [[element structure of symmetric group:S6]]
	\|-
	\| {{arithmetic function value given order\|number of conjugacy classes\|11\|720}} \|\| As <math>\! ~~S_k~~, k = 6:</math> the number of conjugacy classes is <math>p(k) = p(6) = 11</math>, where <math>p</math> is the [[number of unordered integer partitions]]; see [[cycle type determines conjugacy class]].
	\|-		\|-
	\| {{arithmetic function value given order\|number of conjugacy classes of subgroups\|56\|720}} \|\|		\| {{arithmetic function value given order\|number of conjugacy classes of subgroups\|56\|720}} \|\|
			\|-
			\| {{arithmetic function value given order\|number of subgroups\|1455\|720}} \|\|
	\|}		\|}

Vipul at 02:08, 5 December 2011

2011-12-05T02:08:37Z

← Older revision		Revision as of 02:08, 5 December 2011
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	{{particular group}}		{{particular group}}
	[[importance rank::3\| ]]		[[importance rank::2\| ]]
	==Definition==		==Definition==

Vipul at 02:03, 5 December 2011

2011-12-05T02:03:39Z

← Older revision		Revision as of 02:03, 5 December 2011
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	{{particular group}}		{{particular group}}
			[[importance rank::3\| ]]
	==Definition==		==Definition==

Vipul at 22:38, 14 May 2011

2011-05-14T22:38:54Z

← Older revision		Revision as of 22:38, 14 May 2011
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	\|}		\|}

			==Group properties==

			{\| class="sortable" border="1"
			! Property !! Satisfied? !! Explanation
			\|-
			\| [[dissatisfies property::abelian group]] \|\| No \|\| <math>(1,2)</math> and <math>(2,3)</math> don't commute. In fact, <math>S_n</math> is non-abelian for <math>n \ge 3</math>.
			\|-
			\| [[dissatisfies property::nilpotent group]] \|\| No \|\|
			\|-
			\| [[dissatisfies property::solvable group]] \|\| No \|\|
			\|-
			\| [[dissatisfies property::simple group]] \|\| No \|\| has [[alternating group:A6]] as a proper nontrivial normal subgroup
			\|-
			\| [[satisfies property::almost simple group]] \|\| Yes \|\| sandwiched between the simple group [[alternating group:A6]] and the automorphism group thereof. See also [[symmetric groups are almost simple]].
			\|-
			\| [[dissatisfies property::quasisimple group]] \|\| No \|\|
			\|-
			\| [[satisfies property::one-headed group]] \|\| Yes \|\| [[alternating group:A6]] is the unique [[maximal normal subgroup]]
			\|-
			\| [[satisfies property::monolithic group]] \|\| Yes \|\| [[alternating group:A6]] is the unique [[minimal normal subgroup]]
			\|-
			\| [[satisfies property::T-group]] \|\| Yes \|\|
			\|-
			\| [[satisfies property::rational representation group]] \|\| Yes \|\| See [[symmetric groups are rational representation]], [[linear representation theory of symmetric groups]]
			\|-
			\| [[satisfies property::rational group]] \|\| Yes \|\| See [[symmetric groups are rational]]
			\|}
	==Elements==		==Elements==