T-group - Revision history

Vipul: /* Definition */

2013-01-12T18:28:22Z

Definition

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	\| 3 \|\| normal equals transitively normal \|\| any [[normal subgroup]] is [[defining ingredient::transitively normal subgroup\|transitively normal]] \|\| for any normal subgroup <math>K</math> of <math>G</math>, it is true that any normal subgroup of <math>K</math> is normal in <math>G</math>.		\| 3 \|\| normal equals transitively normal \|\| any [[normal subgroup]] is [[defining ingredient::transitively normal subgroup\|transitively normal]] \|\| for any normal subgroup <math>K</math> of <math>G</math>, it is true that any normal subgroup of <math>K</math> is normal in <math>G</math>.
	\|}		\|}

			{{tabular definition format}}

	==Examples==		==Examples==

Vipul at 18:17, 12 January 2013

2013-01-12T18:17:40Z

@@ Line 13: / Line 13: @@
 ==Definition==
-===Symbol-free definition===
+A [[group]] is termed a '''T-group''' if it satisfies the following equivalent conditions:
-A [[group]] is termed a '''T-group''' if it satisfies the following equivalent conditions:
+===Extreme examples===
-* Any [[defining ingredient::subnormal subgroup]] of the group is [[defining ingredient::normal subgroup|normal]] in the group.
+===Groups dissatisfying the property===
-===Definition with symbols===
+{{groups dissatisfying property sorted by importance rank}}
-A group <math>G</math> is termed a '''T-group''' if whenever <math>H</math> is normal in <math>G</math> and <math>K</math> is normal in <math>H</math>, <math>K</math> is also normal in <math>G</math>.
+==Metaproperties==
 ==Formalisms==
 {{subgroup property collapse|normal subgroup|transitively normal subgroup}}
@@ Line 39: / Line 64: @@
 ===Stronger properties===
-* [[Abelian group]]
+{| class="sortable" border="1"
-* [[Dedekind group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[T*-group]]
+|-
-* [[Simple group]]
+| [[Weaker than::Dedekind group]] || every subgroup is [[normal subgroup|normal]] || (direct from definition) || simple non-abelian groups provide counterexamples || {{intermediate notions short|T-group|Dedekind group}}
 ===Conjunction with other properties===
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 * [[HN-group]] (when we are working with [[finite group]]s)
 ==Effect of property operators==

Vipul: /* Effect of property operators */

2008-11-22T23:37:20Z

Effect of property operators

← Older revision		Revision as of 23:37, 22 November 2008
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	The property of being a group such that ''every'' subgroup of it is a T-group, is termed the property of being a [[T*-group]].		The property of being a group such that ''every'' subgroup of it is a T-group, is termed the property of being a [[T*-group]].

			==Testing==

			{{GAP code for group property\|
			test = IsTGroup}}

			There is no built-in GAP command to check whether a group is a T-group, but a short snippet of code, available at [[GAP:IsTGroup]], can achieve this.

	== References ==		== References ==

Vipul: /* Formalisms */

2008-08-21T16:01:21Z

Formalisms

← Older revision		Revision as of 16:01, 21 August 2008
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	==Formalisms==		==Formalisms==
	{{subgroup property collapse}}		{{subgroup property collapse\|normal subgroup\|transitively normal subgroup}}

	The property of being a T-group can be viewed as any of these subgroup property collapses:		The property of being a T-group can be viewed as any of these subgroup property collapses:

	* The collapse [[transitively normal subgroup]] = [[normal subgroup]]		* The collapse [[transitively normal subgroup]] = [[normal subgroup]]
	* The collapse [[normal subgroup]] = [[subnormal subgroup]]		* The collapse [[normal subgroup]] = [[defining ingredient::subnormal subgroup]]

	{{obtainedbyapplyingthe\|transitivity-forcing operator\|normal subgroup}}		{{obtainedbyapplyingthe\|transitivity-forcing operator\|normal subgroup}}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* References */

2008-05-23T16:25:09Z

References

← Older revision		Revision as of 16:25, 23 May 2008
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	===Textbook references===		===Textbook references===

	* {{booklink\|RobinsonGT}}, Page 402-405, ''Section 13.4: Groups in which normality is a transitive relation''		* {{booklink-defined\|RobinsonGT}}, Page 402-405, ''Section 13.4: Groups in which normality is a transitive relation''

	===Journal references===		===Journal references===

Vipul: /* Definition */

2008-05-09T17:02:19Z

Definition

← Older revision		Revision as of 17:02, 9 May 2008
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	===Symbol-free definition===		===Symbol-free definition===

	A [[group]] is termed a '''T-group''' if ~~any~~ [[subnormal subgroup]] of the group is [[normal subgroup\|normal]] in the group. ~~In other words, a T group is~~ a ~~group such that any~~ normal subgroup of a normal subgroup of it is normal in it.		A [[group]] is termed a '''T-group''' if it satisfies the following equivalent conditions:

			* Any [[defining ingredient::subnormal subgroup]] of the group is [[defining ingredient::normal subgroup\|normal]] in the group.
			* Any [[defining ingredient::2-subnormal subgroup]] (i.e. any normal subgroup of a normal subgroup) of a normal subgroup of it is normal in it
			* Any [[normal subgroup]] is [[defining ingredient::transitively normal subgroup\|transitively normal]]

	===Definition with symbols===		===Definition with symbols===

Vipul: 20 revisions

2008-05-08T00:25:50Z

20 revisions

← Older revision	Revision as of 00:25, 8 May 2008
(No difference)

Vipul: /* Metaproperties */

2008-03-29T14:04:11Z

Metaproperties

← Older revision		Revision as of 14:04, 29 March 2008
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	==Metaproperties==		==Metaproperties==

			{{not S-closed}}

			A subgroup of a T-group need not be a T-group. A group in which every subgroup is a T-group, is termed a [[T*-group]].

	{{Q-closed}}		{{Q-closed}}

	Any quotient of a T-group is a T-group. This follows from the fact that a subgroup in the quotient is normal if and only if its full inverse image is normal in the original group.		Any quotient of a T-group is a T-group. This follows from the fact that a subgroup in the quotient is normal if and only if its full inverse image is normal in the original group.

			{{normal subgroup-closed}}

			Any [[normal subgroup]] (and more generally, any [[subnormal subgroup]]) of a T-group is a T-group. {{further\|[[T is normal subgroup-closed]]}}

	==Effect of property operators==		==Effect of property operators==

Vipul: /* References */

2008-03-01T15:26:54Z

References

← Older revision		Revision as of 15:26, 1 March 2008
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	===Textbook references===		===Textbook references===

	* {{booklink\|RobinsonGT}}, Page 402-405		* {{booklink\|RobinsonGT}}, Page 402-405, ''Section 13.4: Groups in which normality is a transitive relation''

	===Journal references===		===Journal references===

Vipul: /* References */

2008-03-01T15:26:19Z

References

← Older revision		Revision as of 15:26, 1 March 2008
Line 63:		Line 63:
	===Textbook references===		===Textbook references===

	* {{RobinsonGT}}, Page 402-405		* {{booklink\|RobinsonGT}}, Page 402-405

	===Journal references===		===Journal references===