Baer correspondence - Revision history

Vipul: /* Interpretation in terms of splitting of short exact sequence */

2017-07-17T15:51:15Z

Interpretation in terms of splitting of short exact sequence

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	===Interpretation in terms of splitting of short exact sequence===		===Interpretation in terms of splitting of short exact sequence===

	[[~~Interpretation~~ of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem]]		See [[interpretation of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem]].

	== Induced correspondences on subgroups ==		== Induced correspondences on subgroups ==

Vipul at 15:50, 17 July 2017

2017-07-17T15:50:51Z

← Older revision		Revision as of 15:50, 17 July 2017
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	This is a special case of the [[twisted multiplication of a 2-powered group]].		This is a special case of the [[twisted multiplication of a 2-powered group]].

			===Interpretation in terms of splitting of short exact sequence===

			[[Interpretation of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem]]

	== Induced correspondences on subgroups ==		== Induced correspondences on subgroups ==

Vipul: /* Correspondence between characteristic subgroups and characteristic subrings (the same as characteristic ideals) */

2017-07-03T02:15:54Z

Correspondence between characteristic subgroups and characteristic subrings (the same as characteristic ideals)

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	The Baer correspondence gives a correspondence:		The Baer correspondence gives a correspondence:

	[[Characteristic subgroup\|Characteristic]] 2-powered subgroups of <math>G</math> <math>\leftrightarrow</math> [[Characteristic subring\|Characteristic]] 2-powered subrings of <math>L</math> = [[Characteristic ideal\|Characteristic]] 2-powered ideals of <math>L</math>		[[Characteristic subgroup\|Characteristic]] 2-powered subgroups of <math>G</math> <math>\leftrightarrow</math> [[Characteristic Lie subring\|Characteristic]] 2-powered subrings of <math>L</math> = [[Characteristic ideal\|Characteristic]] 2-powered ideals of <math>L</math>

	Since [[characteristic implies normal]] on the group side, we can equate characteristic 2-powered subrings and characteristic 2-powered ideals on the Lie ring side. Note that, for Lie rings in general, [[characteristic not implies ideal]], but the collapse in this case follows from the Baer correspondence.		Since [[characteristic implies normal]] on the group side, we can equate characteristic 2-powered subrings and characteristic 2-powered ideals on the Lie ring side. Note that, for Lie rings in general, [[characteristic not implies ideal]], but the collapse in this case follows from the Baer correspondence.

Vipul: /* Correspondence between twisted subgroups and subgroups of additive group of Lie ring */

2017-07-03T02:15:44Z

Correspondence between twisted subgroups and subgroups of additive group of Lie ring

← Older revision		Revision as of 02:15, 3 July 2017
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	2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>		2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>

	The correspondence arises as a result of the [[~~#Twisted product\|~~twisted ~~product discussed earlier on this page~~]].		The correspondence arises as a result of the [[twisted multiplication of a 2-powered group]].

	==Examples==		==Examples==

Vipul at 02:15, 3 July 2017

2017-07-03T02:15:06Z

← Older revision		Revision as of 02:15, 3 July 2017
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	2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>		2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>

	The correspondence arises as a result of the [[#~~twisted~~ product\|twisted product discussed earlier on this page]].		The correspondence arises as a result of the [[#Twisted product\|twisted product discussed earlier on this page]].

	==Examples==		==Examples==

Vipul: /* Induced correspondences on subgroups */

2017-07-03T02:14:39Z

Induced correspondences on subgroups

← Older revision		Revision as of 02:14, 3 July 2017
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	Among the generalizations discussed at [[generalized Baer correspondence]], only the LUCS-Baer correspondence and the [[UCS-Baer correspondence]] improve generality when already restricted to subgroups of Baer Lie groups. The UCS-Baer correspondence is intermediate between the Baer correspondence and the LUCS-Baer correspondence, and therefore offers an intermediate level of coverage. The other generalizations primarily add generality by covering cases where the group has 2-torsion, but this will not occur inside subgroups of Baer Lie groups.		Among the generalizations discussed at [[generalized Baer correspondence]], only the LUCS-Baer correspondence and the [[UCS-Baer correspondence]] improve generality when already restricted to subgroups of Baer Lie groups. The UCS-Baer correspondence is intermediate between the Baer correspondence and the LUCS-Baer correspondence, and therefore offers an intermediate level of coverage. The other generalizations primarily add generality by covering cases where the group has 2-torsion, but this will not occur inside subgroups of Baer Lie groups.

			In the special case that <math>G</math> is finite, or even periodic (every element has finite order), the Baer correspondence (or any of its generalizations) covers all subgroups and all subrings, and we get:

			Subgroups of <math>G</math> <math>\leftrightarrow</math> Subrings of <math>L</math>

			=== Correspondence between normal subgroups and ideals ===

			The correspondence between 2-powered subgroups and 2-powered subrings can be restricted to the case of normal subgroups and ideals. We get:

			2-powered normal subgroups of <math>G</math> <math>\leftrightarrow</math> 2-powered ideals of <math>L</math>

			The LUCS-Baer correspondence generalization helps us cover more subgroups and subrings, and we get a larger correspondence:

			Normal subgroups where every element of its derived subgroup has a (unique) square root in its center (LUCS-Baer Lie subgroups) <math>\leftrightarrow</math> Ideals where every element of the derived subring has a (unique) half in its center (LUCS-Baer Lie subrings)

			In particular, this generalization covers both the case of 2-powered normal subgroups (respectively, 2-powered ideals) and the case of abelian normal subgroups (respectively, abelian ideals).

			In the special case that <math>G</math> is finite, or even periodic (every element has finite order), the Baer correspondence (or any of its generalizations) covers all subgroups and all subrings, and we get:

			Normal subgroups of <math>G</math> <math>\leftrightarrow</math> Ideals of <math>L</math>

			=== Correspondence between characteristic subgroups and characteristic subrings (the same as characteristic ideals) ===

			The Baer correspondence gives a correspondence:

			[[Characteristic subgroup\|Characteristic]] 2-powered subgroups of <math>G</math> <math>\leftrightarrow</math> [[Characteristic subring\|Characteristic]] 2-powered subrings of <math>L</math> = [[Characteristic ideal\|Characteristic]] 2-powered ideals of <math>L</math>

			Since [[characteristic implies normal]] on the group side, we can equate characteristic 2-powered subrings and characteristic 2-powered ideals on the Lie ring side. Note that, for Lie rings in general, [[characteristic not implies ideal]], but the collapse in this case follows from the Baer correspondence.

			In the special case that <math>G</math> is finite, or even periodic (every element has finite order), the Baer correspondence (or any of its generalizations) covers all subgroups and all subrings, and we get:

			Characteristic subgroups of <math>G</math> <math>\leftrightarrow</math> Characteristic subrings of <math>L</math> = Characteristic ideals of <math>L</math>

	=== Correspondence between twisted subgroups and subgroups of additive group of Lie ring ===		=== Correspondence between twisted subgroups and subgroups of additive group of Lie ring ===

Vipul: /* Correspondence between twisted subgroups and subgroups of additive group of Lie ring */

2017-07-03T02:05:22Z

Correspondence between twisted subgroups and subgroups of additive group of Lie ring

← Older revision		Revision as of 02:05, 3 July 2017
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	2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>		2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>

			The correspondence arises as a result of the [[#twisted product\|twisted product discussed earlier on this page]].

	==Examples==		==Examples==

Vipul: /* Correspondence between twisted subgroups and subgroups of additive group of Lie ring */

2017-07-03T02:03:42Z

Correspondence between twisted subgroups and subgroups of additive group of Lie ring

← Older revision		Revision as of 02:03, 3 July 2017
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	The Baer correspondence induces a correspondence:		The Baer correspondence induces a correspondence:

	2-powered twisted ~~subgroups~~ of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>		2-powered [[twisted subgroup]]s of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>

	==Examples==		==Examples==

Vipul: /* Correspondence between twisted subgroups and subgroups of additive group of Lie ring */

2017-07-03T02:03:33Z

Correspondence between twisted subgroups and subgroups of additive group of Lie ring

← Older revision		Revision as of 02:03, 3 July 2017
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	=== Correspondence between twisted subgroups and subgroups of additive group of Lie ring ===		=== Correspondence between twisted subgroups and subgroups of additive group of Lie ring ===

	{{further\|Baer correspondence establishes a correspondence between 2-powered twisted subgroups of the group and 2-powered subgroups of additive group of the Lie ring}}		{{further\|[[Baer correspondence establishes a correspondence between 2-powered twisted subgroups of the group and 2-powered subgroups of additive group of the Lie ring]]}}

	The Baer correspondence induces a correspondence:		The Baer correspondence induces a correspondence:

Vipul at 02:02, 3 July 2017

2017-07-03T02:02:00Z

← Older revision		Revision as of 02:02, 3 July 2017
Line 123:		Line 123:

	This is a special case of the [[twisted multiplication of a 2-powered group]].		This is a special case of the [[twisted multiplication of a 2-powered group]].

			== Induced correspondences on subgroups ==

			Consider a Baer Lie group <math>G</math> and corresponding Baer Lie ring <math>L</math>.

			=== Correspondence between subgroups and subrings ===

			In general, [[Baer Lie property is not subgroup-closed]], because the condition of being 2-powered (i.e., uniquely 2-divisible) is not inherited by subgroups. However, by restricting to the 2-powered case, the Baer correspondence gives a correspondence:

			2-powered subgroups (Baer Lie subgroups) of <math>G</math> <math>\leftrightarrow</math> 2-powered subrings (Baer Lie subrings) of <math>L</math>

			Specifically, a subset that is a 2-powered subgroup under the group structure is a 2-powered subring under the Lie ring structure.

			A slightly more general correspondence is obtained by noting that the Baer correspondence generalizes to the [[LUCS-Baer correspondence]], so we get:

			Subgroups where every element of its derived subgroup has a (unique) square root in its center (LUCS-Baer Lie subgroups) <math>\leftrightarrow</math> Subrings where every element of the derived subring has a (unique) half in its center (LUCS-Baer Lie subrings)

			This generalized correspondence includes the correspondence between Baer Lie subgroups and Baer Lie subrings, and also includes the correspondence between abelian subgroups and abelian Lie subrings.

			Among the generalizations discussed at [[generalized Baer correspondence]], only the LUCS-Baer correspondence and the [[UCS-Baer correspondence]] improve generality when already restricted to subgroups of Baer Lie groups. The UCS-Baer correspondence is intermediate between the Baer correspondence and the LUCS-Baer correspondence, and therefore offers an intermediate level of coverage. The other generalizations primarily add generality by covering cases where the group has 2-torsion, but this will not occur inside subgroups of Baer Lie groups.

			=== Correspondence between twisted subgroups and subgroups of additive group of Lie ring ===

			{{further\|Baer correspondence establishes a correspondence between 2-powered twisted subgroups of the group and 2-powered subgroups of additive group of the Lie ring}}

			The Baer correspondence induces a correspondence:

			2-powered twisted subgroups of <math>G</math><math>\leftrightarrow</math> 2-powered additive subgroups of <math>L</math>

	==Examples==		==Examples==