Normal subloop - Revision history

Vipul: /* Facts */

2011-08-21T05:59:02Z

Facts

← Older revision		Revision as of 05:59, 21 August 2011
Line 30:		Line 30:
	* Given a normal subloop, the left multiplication group corresponding to that subloop, is a [[normal subgroup]] of the left multiplication group corresponding to the whole algebra loop. Notice that for this, we crucially need the equality of all three parts: <math>a * (b * N) = (a * b) * N = a * (N * b)</math>.		* Given a normal subloop, the left multiplication group corresponding to that subloop, is a [[normal subgroup]] of the left multiplication group corresponding to the whole algebra loop. Notice that for this, we crucially need the equality of all three parts: <math>a * (b * N) = (a * b) * N = a * (N * b)</math>.
	* Further, the left multiplication group of the quotient loop equals the quotient of the left multiplication group of the whole loop, by that of the subgroup.		* Further, the left multiplication group of the quotient loop equals the quotient of the left multiplication group of the whole loop, by that of the subgroup.

			==Relation with other properties==

			===Stronger properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::direct factor of a loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|direct factor of a loop}}
			\|-
			\| [[Weaker than::central factor of a loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}
			\|-
			\| [[Weaker than::central subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central subloop}}
			\|-
			\| [[Weaker than::cocentral subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|cocentral subloop}}
			\|}

			===Weaker properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::subnormal subloop]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::2-subnormal subloop]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Lagrange-like subloop]] (in a [[finite loop]]) \|\| \|\| \|\| \|\|
			\|}

Vipul at 06:40, 20 August 2011

2011-08-20T06:40:16Z

← Older revision		Revision as of 06:40, 20 August 2011
Line 5:		Line 5:
	old specific context = subgroup\|		old specific context = subgroup\|
	old property = normal subgroup\|		old property = normal subgroup\|
	new generic context = ~~algebra~~ loop\|		new generic context = loop\|
	new specific context = subloop}}		new specific context = subloop}}

Vipul at 20:01, 24 August 2008

2008-08-24T20:01:46Z

← Older revision		Revision as of 20:01, 24 August 2008
Line 1:		Line 1:
	{{subloop property}}		{{subloop property}}

	{{analogue ~~in-~~of\|algebra loop\|~~subgroup\|normality~~}}		{{analogue of property\|
			old generic context = group\|
			old specific context = subgroup\|
			old property = normal subgroup\|
			new generic context = algebra loop\|
			new specific context = subloop}}

	==Definition==		==Definition==

Vipul: 4 revisions

2008-05-07T23:57:07Z

4 revisions

← Older revision	Revision as of 23:57, 7 May 2008
(No difference)

Vipul at 00:00, 27 April 2008

2008-04-27T00:00:07Z

← Older revision		Revision as of 00:00, 27 April 2008
Line 1:		Line 1:
	{{subloop property}}		{{subloop property}}

	{{~~analog~~ in-of\|algebra loop\|subgroup\|normality}}		{{analogue in-of\|algebra loop\|subgroup\|normality}}

	==Definition==		==Definition==

Vipul at 23:25, 10 March 2008

2008-03-10T23:25:54Z

← Older revision		Revision as of 23:25, 10 March 2008
Line 1:		Line 1:
	{{subloop property}}		{{subloop property}}

	{{~~analogue~~ in-of\|algebra loop\|subgroup\|normality}}		{{analog in-of\|algebra loop\|subgroup\|normality}}

	==Definition==		==Definition==

Vipul at 09:18, 13 June 2007

2007-06-13T09:18:17Z

@@ Line 9: / Line 9: @@
 A [[subloop]] <math>N</math> of an [[algebra loop]] <math>L</math> is said to be '''normal''' if, for any <math>a,b \in L</math>, the following holds:
-<math>(ab)N = a(bN) = a(Nb)</math>
+<math>(a * b) * N = a * (b * N) = a * (N * b)</math>
-Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be [[assiciative binary operation|associative]].
+Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be [[associative binary operation|associative]].

Vipul at 08:20, 13 June 2007

2007-06-13T08:20:58Z

New page

{{subloop property}}

{{analogue in-of|algebra loop|subgroup|normality}}

==Definition==

===Definition with symbols===

A [[subloop]] <math>N</math> of an [[algebra loop]] <math>L</math> is said to be '''normal''' if, for any <math>a,b \in L</math>, the following holds:

<math>(ab)N = a(bN) = a(Nb)</math>

Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be [[assiciative binary operation|associative]].