Center - Revision history

Vipul: /* Properties not satisfied */

2016-06-01T20:14:28Z

Properties not satisfied

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	\| [[Dissatisfies property::normality-preserving endomorphism-invariant subgroup]] \|\| invariant under any [[normality-preserving endomorphism]] \|\| [[center not is normality-preserving endomorphism-invariant]]		\| [[Dissatisfies property::normality-preserving endomorphism-invariant subgroup]] \|\| invariant under any [[normality-preserving endomorphism]] \|\| [[center not is normality-preserving endomorphism-invariant]]
	\|-		\|-
	\| [[Dissatisfies property::divisibility-closed subgroup]] \|\| if <math>n</math> is a positive integer such that every element of the group has a <math>n^{th}</math> root, then every element of the subgroup also has a <math>n^{th}</math> root in the subgroup. \|\| [[center not is divisibility-closed]]		\| [[Dissatisfies property::divisibility-closed subgroup]] \|\| if <math>n</math> is a positive integer such that every element of the group has a <math>n^{th}</math> root, then every element of the subgroup also has a <math>n^{th}</math> root in the subgroup. \|\| [[center not is divisibility-closed]] (however, [[upper central series members are completely divisibility-closed in nilpotent group]])
	\|-		\|-
	\| [[Dissatisfies property::endomorphism kernel]] \|\| kernel of an endomorphism of the group. \|\| [[center not is endomorphism kernel]]		\| [[Dissatisfies property::endomorphism kernel]] \|\| kernel of an endomorphism of the group. \|\| [[center not is endomorphism kernel]]

Vipul: /* Smaller subgroup-defining functions */

2014-02-28T17:30:10Z

Smaller subgroup-defining functions

← Older revision		Revision as of 17:30, 28 February 2014
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	* [[Contains::Absolute center]]: This is the set of elements of the group fixed by ''every'' [[automorphism]] (not just by every [[inner automorphism]]).		* [[Contains::Absolute center]]: This is the set of elements of the group fixed by ''every'' [[automorphism]] (not just by every [[inner automorphism]]).
	* [[Contains::Epicenter]]		* [[Contains::Epicenter]]: Intersection of inverse images of centers for all [[central extension]]s.
	* For a [[group of prime power order]], the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the [[socle]] of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts. {{further\|[[socle equals Omega-1 of center in nilpotent p-group]]}}		* For a [[group of prime power order]], the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the [[socle]] of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts. {{further\|[[socle equals Omega-1 of center in nilpotent p-group]]}}

Vipul: /* Smaller subgroup-defining functions */

2014-02-28T17:10:42Z

Smaller subgroup-defining functions

← Older revision		Revision as of 17:10, 28 February 2014
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	===Smaller subgroup-defining functions===		===Smaller subgroup-defining functions===

			* [[Contains::Absolute center]]: This is the set of elements of the group fixed by ''every'' [[automorphism]] (not just by every [[inner automorphism]]).
	* [[Contains::Epicenter]]		* [[Contains::Epicenter]]
	* For a [[group of prime power order]], the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the [[socle]] of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts. {{further\|[[socle equals Omega-1 of center in nilpotent p-group]]}}		* For a [[group of prime power order]], the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the [[socle]] of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted <math>\Omega_1(Z(G))</math> where <math>G</math> is the whole group, is important in many contexts. {{further\|[[socle equals Omega-1 of center in nilpotent p-group]]}}

Vipul: /* Subgroup-defining function properties */

2013-07-27T17:06:10Z

Subgroup-defining function properties

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	==Subgroup-defining function properties==		==Subgroup-defining function properties==

	{~~{reverse monotone sdf}}~~		{\| class="sortable" border="1"
			! Property name !! Satisfied? !! Proof !! Statement with symbols
	~~The '''center'''~~ [[subgroup-defining function]] ~~is reverse monotone. That is:~~		\|-
			\| [[satisfies property::reverse monotone subgroup-defining function]] \|\| Yes \|\| \|\| Suppose <math>H \le G</math>. Then, <math>H \cap Z(G) \le Z(H)</math>.
	~~Let~~ <math>H \le G</math> ~~be groups~~. Then, <math>Z(H)</math> ~~contains the~~ group <math>Z(G) ~~\cap H~~</math>.		\|-
			\| [[satisfies property::idempotent subgroup-defining function]] \|\| Yes \|\| \|\| For any group <math>G</math>, <math>Z(Z(G)) = Z(G)</math>, i.e., the center of the center is the center.
	~~{{idempotent sdf}}~~		\|}

	~~The~~ center of the center is the center. ~~This is because the center is an [[Abelian group]], and the center of any Abelian group is itself.~~

	==In groups with additional structure==		==In groups with additional structure==

Vipul: /* Effect of operators */

2013-07-27T17:02:55Z

Effect of operators

← Older revision		Revision as of 17:02, 27 July 2013
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	==Effect of operators==		==Effect of operators==

	==~~=Fixed~~-point operator~~===~~		{\| class="sortable" border="1"
			! Operator !! Meaning of application to center !! Value
	A group equals its own center ~~if and only if it is an~~ [[~~Abelian~~ group]].		\|-
			\| fixed-point operator \|\| a group that equals its own center \|\| [[abelian group]]
	~~===Free~~ operator~~===~~		\|-
			\| free operator \|\| a group whose center is trivial \|\| [[centerless group]]
	A group whose center is trivial ~~is termed a~~ [[centerless group]].		\|}

	==Subgroup-defining function properties==		==Subgroup-defining function properties==

Vipul: /* Properties satisfied */

2013-07-27T16:59:02Z

Properties satisfied

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	\|-		\|-
	\| [[Satisfies property::marginal subgroup]] \|\| there is a set of words such that the center is precisely the set of elements by which left or right multiplication on any letter of the word does not affect the value (the set of words here is the singleton set comprising the [[commutator]] word) \|\| [[center is marginal]]		\| [[Satisfies property::marginal subgroup]] \|\| there is a set of words such that the center is precisely the set of elements by which left or right multiplication on any letter of the word does not affect the value (the set of words here is the singleton set comprising the [[commutator]] word) \|\| [[center is marginal]]
			\|-
			\| [[Satisfies property::marginal subgroup of finite type]] \|\|same definition as for marginal subgroup, but we insist that the set of words be finite (satisfied by the center because we can use a ''single'' word). \|\| [[center is marginal of finite type]]
	\|-		\|-
	\| [[Satisfies property::finite direct power-closed characteristic subgroup]] \|\| in any finite direct power of the whole group, the corresponding direct power of the center is a [[characteristic subgroup]] \|\| [[center is finite direct power-closed characteristic]]		\| [[Satisfies property::finite direct power-closed characteristic subgroup]] \|\| in any finite direct power of the whole group, the corresponding direct power of the center is a [[characteristic subgroup]] \|\| [[center is finite direct power-closed characteristic]]

Vipul: /* Associated constructions */

2013-07-27T16:56:37Z

Associated constructions

← Older revision		Revision as of 16:56, 27 July 2013
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	==Associated constructions==		==Associated constructions==
	~~{{associated qdf\|Inner automorphism group}}~~

	The quotient of a group by its center is isomorphic to the group of [[inner automorphism]]s, This is because the map ~~fro ma~~ group to its automorphism group that sends <math>g</math> to <math>c_g: x \mapsto gxg^{-1}</math> is a homomorphism, and its kernel is precisely the center <math>Z(G)</math>.		{\| class="sortable" border="1"
			! Construction !! Name !! Description
	{{associated as\|~~Upper~~ central series}}		\|-
			\| associated [[quotient-defining function]] \|\| [[inner automorphism group]] \|\| The quotient of a group by its center is isomorphic to the group of [[inner automorphism]]s, i.e. the subgroup of the [[automorphism group]] comprising those automorphisms that can be expressed using conjugation maps. This is because the map from a group to its automorphism group that sends <math>g</math> to <math>c_g: x \mapsto gxg^{-1}</math> is a homomorphism, and its kernel is precisely the center <math>Z(G)</math>.
	Start with a group <math>G</math>. The upper central series of <math>G</math> is defined as follows. Consider <math>Z^1(G) = Z(G)</math>. Let <math>Z^i(G)</math>, in general, be the inverse image in <math>G</math> of <math>Z(G/Z^{i-1}(G))</math> under the canonical projection <math>G \to G/Z^{i-1}(G)</math>. Essentially we are iterating the [[quotient-defining function]] that sends a group to the inner automorphism group, and taking the kernel at each step. However, we are pulling back that kernel all the way to <math>G</math>.		\|-
			\| associated [[ascending series]] \|\| [[upper central series]] \|\| Start with a group <math>G</math>. The upper central series of <math>G</math> is defined as follows. Consider <math>Z^1(G) = Z(G)</math>. Let <math>Z^i(G)</math>, in general, be the inverse image in <math>G</math> of <math>Z(G/Z^{i-1}(G))</math> under the canonical projection <math>G \to G/Z^{i-1}(G)</math>. Essentially we are iterating the [[quotient-defining function]] that sends a group to the inner automorphism group, and taking the kernel at each step. However, we are pulling back that kernel all the way to <math>G</math>. <br>By convention (and commonsense) <math>Z^0(G)</math> is the [[trivial group]].<br> A group for which the upper central series terminates in finite length at the whole group is termed a [[nilpotent group]].
	By convention (and commonsense) <math>Z^0(G)</math> is the [[trivial group]].		\|}

	A group for which the upper central series terminates in finite length at the whole group is termed a [[nilpotent group]].

	==Examples==		==Examples==

Vipul: /* Properties not satisfied */

2013-07-01T21:57:26Z

Properties not satisfied

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	\| [[Dissatisfies property::endomorphism kernel]] \|\| kernel of an endomorphism of the group. \|\| [[center not is endomorphism kernel]]		\| [[Dissatisfies property::endomorphism kernel]] \|\| kernel of an endomorphism of the group. \|\| [[center not is endomorphism kernel]]
	\|-		\|-
	\| [[dissatisfies property::quotient-local powering-invariant subgroup]] \|\| If an element of the group has a unique <math>n^{th}</math> root for some <math>n</math>, then its image modulo the subgroup in question should also have a unique <math>n^{th}</math> root. \|\| [[center not is quotient-local powering-invariant]]		\| [[dissatisfies property::quotient-local powering-invariant subgroup]] \|\| If an element of the group has a unique <math>n^{th}</math> root for some <math>n</math>, then its image modulo the subgroup in question should also have a unique <math>n^{th}</math> root. \|\| [[center not is quotient-local powering-invariant]] (however, [[upper central series members are quotient-local powering-invariant in nilpotent group]])
	\|-		\|-
	\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant]]		\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant]] (however, [[upper central series members are intermediately powering-invariant in nilpotent group]])
	\|-		\|-
	\| [[dissatisfies property::intermediately local powering-invariant subgroup]] \|\| [[local powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately local powering-invariant]]		\| [[dissatisfies property::intermediately local powering-invariant subgroup]] \|\| [[local powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately local powering-invariant]] (however, [[upper central series members are intermediately local powering-invariant in nilpotent group]])
	\|}		\|}

Vipul: /* Properties not satisfied */

2013-07-01T21:51:12Z

Properties not satisfied

← Older revision		Revision as of 21:51, 1 July 2013
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	\|-		\|-
	\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant]]		\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant]]
			\|-
			\| [[dissatisfies property::intermediately local powering-invariant subgroup]] \|\| [[local powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately local powering-invariant]]
	\|}		\|}

Vipul: /* Properties not satisfied */

2013-07-01T21:45:46Z

Properties not satisfied

← Older revision		Revision as of 21:45, 1 July 2013
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	\| [[dissatisfies property::quotient-local powering-invariant subgroup]] \|\| If an element of the group has a unique <math>n^{th}</math> root for some <math>n</math>, then its image modulo the subgroup in question should also have a unique <math>n^{th}</math> root. \|\| [[center not is quotient-local powering-invariant]]		\| [[dissatisfies property::quotient-local powering-invariant subgroup]] \|\| If an element of the group has a unique <math>n^{th}</math> root for some <math>n</math>, then its image modulo the subgroup in question should also have a unique <math>n^{th}</math> root. \|\| [[center not is quotient-local powering-invariant]]
	\|-		\|-
	\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant ~~in solvable group~~]]		\| [[dissatisfies property::intermediately powering-invariant subgroup]] \|\| [[powering-invariant subgroup]] inside every intermediate subgroup \|\| [[center not is intermediately powering-invariant]]
	\|}		\|}