Normal subloop: Difference between revisions

Latest revision as of 05:59, 21 August 2011

This article defines a property that can be evaluated for a subloop of a loop| View other such properties

ANALOGY: This is an analogue in loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in loops of subgroup properties (OR, View as a tabulated list)

Definition

Definition with symbols

A subloop $N$ of an algebra loop $L$ is said to be normal if, for any $a, b \in L$ , the following holds:

$(a * b) * N = a * (b * N) = a * (N * b)$

Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be associative.

Facts

Quotient by a normal subloop

Given a loop, and a normal subloop, we can define a corresponding quotient loop. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Left multiplication group corresponding to a subloop

The following are true:

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: $a * (b * N) = (a * b) * N = a * (N * b)$ .
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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
direct factor of a loop				\|FULL LIST, MORE INFO
central factor of a loop				\|FULL LIST, MORE INFO
central subloop				\|FULL LIST, MORE INFO
cocentral subloop				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subnormal subloop
2-subnormal subloop
Lagrange-like subloop (in a finite loop)

@@ 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>.
 * 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]]) || || || ||
+|}