Generalized quaternion group:Q16 - Revision history

R-a-jones at 18:38, 19 December 2023

2023-12-19T18:38:19Z

← Older revision		Revision as of 18:38, 19 December 2023
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	[[importance rank::3\| ]]		[[importance rank::3\| ]]
	[[Category:Generalized quaternion groups]]		[[Category:Generalized quaternion groups]]
			[[Category:Dicyclic groups]]
	==Definition==		==Definition==

R-a-jones at 15:49, 15 December 2023

2023-12-15T15:49:46Z

← Older revision		Revision as of 15:49, 15 December 2023
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	{{particular group}}		{{particular group}}
	[[importance rank::3\| ]]		[[importance rank::3\| ]]
			[[Category:Generalized quaternion groups]]
	==Definition==		==Definition==

R-a-jones at 23:32, 26 October 2023

2023-10-26T23:32:21Z

@@ Line 103: / Line 103: @@
 | [[Join of elementary abelian subgroups of maximum order]] || (2) || || [[Subgroup-defining function value::Join of elementary abelian subgroups of maximum order;cyclic group:Z2| ]][[cyclic group:Z2]] ||
 |}
 ==GAP implementation==

R-a-jones: /* Definition */

2023-10-26T23:11:30Z

Definition

← Older revision		Revision as of 23:11, 26 October 2023
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	<math>G := \langle a,b \mid a^4 = b^2 = abab \rangle</math>		<math>G := \langle a,b \mid a^4 = b^2 = abab \rangle</math>

			Equivalently, it is the [[dicyclic group]] of order <math>16</math>.

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

Vipul: /* Subgroups */

2013-01-20T17:15:12Z

Subgroups

← Older revision		Revision as of 17:15, 20 January 2013
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	\|}		\|}

			==Group properties==

			===Important properties===

			{\| class="sortable" border="1"
			! Property !! Satisfied? !! Explanation !! Comment
			\|-
			\| {{group properties because p-group}}
			\|-
			\| [[dissatisfies property::abelian group]] \|\| No \|\| <math>a,b</math> don't commute. \|\|
			\|-
			\| [[satisfies property::metacyclic group]] \|\| Yes \|\| <math>\langle a \rangle</math> is cyclic of order eight, quotient group is cyclic of order two. \|\|
			\|-
			\| [[satisfies property::metabelian group]] \|\| Yes \|\| follows from being metacyclic. \|\|
			\|}

			===Other properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Satisfied? !! Explanation !! Comment
			\|-
			\| [[satisfies property::finite group with periodic cohomology]] \|\| finite group in which every abelian subgroup is cyclic \|\| Yes \|\| \|\|
			\|-
			\| [[satisfies property::Schur-trivial group]] \|\| the [[Schur multiplier]] is trivial \|\| Yes \|\| follows from having periodic cohomology \|\|
			\|-
			\| [[satisfies property::maximal class group]] \|\| [[finite p-group]] of class more than one whose class is one less than the [[prime-base logarithm of order]] \|\| Yes \|\| class is 3, prime-base logarithm of order is 4. \|\|
			\|-
			\| [[satisfies property::UL-equivalent group]] \|\| [[upper central series]] and [[lower central series]] coincide. \|\| \|\|
			\|-
			\| [[satisfies property::stem group]] \|\| the [[center]] is contained in the [[derived subgroup]] \|\| Yes \|\| follows from being a non-abelian UL-equivalent group. \|\|
			\|-
			\| [[satisfies property::directly indecomposable group]] \|\| nontrivial and cannot be expressed as an [[internal direct product]] of nontrivial subgroups \|\| Yes \|\| \|\|
			\|-
			\| [[satisfies property::centrally indecomposable group]] \|\| nontrivial and cannot be expressed as a [[internal central product]] of proper nontrivial subgroups \|\| Yes \|\| \|\|
			\|-
			\| [[satisfies property::splitting-simple group]] \|\| nontrivial and cannot be expressed as an [[internal semidirect product]] of nontrivial subgroups \|\| Yes \|\| \|\|
			\|}
	==Subgroups==		==Subgroups==

Vipul: /* Subgroup-defining functions */

2013-01-20T17:04:42Z

Subgroup-defining functions

← Older revision		Revision as of 17:04, 20 January 2013
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	# Two [[quaternion group]]s of order eight, namely <math>\langle a^2,b \rangle</math> and <math>\langle a^2, ab \rangle</math>. Isomorphic to [[subgroup::quaternion group]]. (2)		# Two [[quaternion group]]s of order eight, namely <math>\langle a^2,b \rangle</math> and <math>\langle a^2, ab \rangle</math>. Isomorphic to [[subgroup::quaternion group]]. (2)
	# The whole group. (1)		# The whole group. (1)

	~~==Subgroup-defining functions==~~

	==Subgroup-defining functions==		==Subgroup-defining functions==

Vipul: /* Arithmetic functions */

2012-05-31T16:42:01Z

Arithmetic functions

@@ Line 19: / Line 19: @@
 degree = 8|
 degree p-log = 3}}
 ==Subgroups==

Vipul: /* GAP implementation */

2012-04-18T04:23:16Z

GAP implementation

← Older revision		Revision as of 04:23, 18 April 2012
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	{{GAP ID\|16\|9}}		{{GAP ID\|16\|9}}
	{{HallSenior\|16\|14}}		{{HallSenior\|16\|14}}

			===Description by presentation===

			<pre>gap> F := FreeGroup(2);
			<free group on the generators [ f1, f2 ]>
			gap> G := F/[F.1^8,F.2^2F.1^(-4),F.2F.1F.2^(-1)F.1];
			<fp group on the generators [ f1, f2 ]>
			gap> IdGroup(G);
			[ 16, 9 ]</pre>

	===Other descriptions===		===Other descriptions===

Vipul: /* GAP implementation */

2012-04-18T04:20:36Z

GAP implementation

← Older revision		Revision as of 04:20, 18 April 2012
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	{{GAP ID\|16\|9}}		{{GAP ID\|16\|9}}
	{{HallSenior\|16\|14}}		{{HallSenior\|16\|14}}

			===Other descriptions===

			{\| class="sortable" border="1"
			! Description !! Functions used
			\|-
			\| <tt>SylowSubgroup(SL(2,7),2)</tt> \|\| [[GAP:SylowSubgroup\|SylowSubgroup]], [[GAP:SL\|SL]]
			\|}

Vipul: /* GAP implementation */

2012-04-18T04:10:15Z

GAP implementation

← Older revision		Revision as of 04:10, 18 April 2012
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	{{GAP ID\|16\|9}}		{{GAP ID\|16\|9}}
			{{HallSenior\|16\|14}}