Subnormal subloop: Difference between revisions

Latest revision as of 10:56, 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: subnormal subgroup
View other analogues of subnormal subgroup | View other analogues in loops of subgroup properties (OR, View as a tabulated list)

Definition

A subloop $S$ of a loop $L$ is termed a subnormal subloop if there exists an ascending chain of subloops:

$S = A_{0} \leq A_{1} \leq \dots \leq A_{n} = L$

such that each $A_{i}$ is a normal subloop of $A_{i + 1}$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subloop				\|FULL LIST, MORE INFO
2-subnormal subloop	normal subloop of normal subloop			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lagrange-like subloop in finite loop				\|FULL LIST, MORE INFO

@@ Line 6: / Line 6: @@
 new specific context = subloop|
 old property = subnormal subgroup}}
+==Definition==
+A [[subloop]] <math>S</math> of a [[loop]] <math>L</math> is termed a '''subnormal subloop''' if there exists an ascending chain of subloops:
+<math>S = A_0 \le A_1 \le \dots \le A_n = L</math>
+such that each <math>A_i</math> is a [[normal subloop]] of <math>A_{i+1}</math>.
+==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::normal subloop]] || || || || {{intermediate notions short|subnormal subloop|normal subloop}}
+|-
+| [[Weaker than::2-subnormal subloop]] || normal subloop of normal subloop || || || {{intermediate notions short|subnormal subloop|2-subnormal subloop}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Lagrange-like subloop]] in finite [[loop]] || || || || {{intermediate notions short|Lagrange-like subloop|subnormal subloop}}
+|}