Lagrange-like subloop: Difference between revisions

Latest revision as of 06:02, 21 August 2011

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

Definition

A subloop of a finite loop is termed a Lagrange-like subloop if the order (i.e., the number of elements) of the subloop divides the order of the loop.

Relation with other properties

Algebra loop satisfying the weak Lagrange property is a finite algebra loop in which every subloop is Lagrange-like.
Algebra loop satisfying the strong Lagrange property is a finite algebra loop in which every subloop satisfies the weak Lagrange property.

Stronger properties

All the properties below are stronger conditional to the ambient loop being finite.

Property	Proof of implication	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
normal subloop	normal implies Lagrange-like	\|FULL LIST, MORE INFO
subnormal subloop		\|FULL LIST, MORE INFO

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[subloop]] of a finite [[algebra loop]] is termed a '''Lagrange-like subloop''' if the order (i.e., the number of elements) of the subloop divides the order of the loop.
+A [[subloop]] of a finite [[loop]] is termed a '''Lagrange-like subloop''' if the order (i.e., the number of elements) of the subloop divides the order of the loop.
 ==Relation with other properties==
@@ Line 9: / Line 9: @@
 * [[Algebra loop satisfying the weak Lagrange property]] is a finite algebra loop in which every subloop is Lagrange-like.
 * [[Algebra loop satisfying the strong Lagrange property]] is a finite algebra loop in which every subloop satisfies the weak Lagrange property.
+===Stronger properties===
+All the properties below are stronger ''conditional'' to the ambient loop being finite.
+{| 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|Lagrange-like subloop|direct factor of a loop}}
+|-
+| [[Weaker than::central factor of a loop]] || || || || {{intermediate notions short|Lagrange-like subloop|central factor of a loop}}
+|-
+| [[Weaker than::central subloop]] || || || || {{intermediate notions short|Lagrange-like subloop|central subloop}}
+|-
+| [[Weaker than::cocentral subloop]] || || || || {{intermediate notions short|Lagrange-like subloop|Lagrange-like subloop|cocentral subloop}}
+|-
+| [[Weaker than::normal subloop]] || || [[normal implies Lagrange-like]] || || {{intermediate notions short|Lagrange-like subloop|Lagrange-like subloop|normal subloop}}
+|-
+| [[Weaker than::subnormal subloop]]|| || || || {{intermediate notions short|Lagrange-like subloop|Lagrange-like subloop|subnormal subloop}}
+|}