Normal complex of groups: Difference between revisions

Latest revision as of 16:16, 25 June 2013

A chain complex of groups $(G_{n},d_{n})_{n\in \mathbb {Z} }$ given as follows:

$\dots \to G_{n}{\stackrel {d_{n}}{\to }}G_{n-1}{\stackrel {d_{n-1}}{\to }}G_{n-2}\to \dots$

is termed a normal complex of groups if, for every $n\in \mathbb {Z}$ , the image group $d_{n}(G_{n})$ is a normal subgroup inside $G_{n-1}$ .

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
chain complex of abelian groups	all the groups are abelian	follows from abelian implies every subgroup is normal		\|FULL LIST, MORE INFO

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
chain complex of groups				\|FULL LIST, MORE INFO

@@ Line 5: / Line 5: @@
 <math> \dots \to G_n \stackrel{d_n}{\to} G_{n-1} \stackrel{d_{n-1}}{\to} G_{n-2} \to \dots</math>
-is termed a '''normal complex of groups''' if, for every <math>n \in \mathbb{Z}</math>, the image group <math>d_n(G_n)</math> is a [defining ingredient::[normal subgroup]] inside <math>G_{n-1}</math>.
+is termed a '''normal complex of groups''' if, for every <math>n \in \mathbb{Z}</math>, the image group <math>d_n(G_n)</math> is a [[defining ingredient::normal subgroup]] inside <math>G_{n-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::chain complex of abelian groups]] || all the groups are abelian || follows from [[abelian implies every subgroup is normal]] || || {{intermediate notions short|normal complex of groups|chain complex of abelian groups}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::chain complex of groups]] || || || || {{intermediate notions short|chain complex of groups|normal complex of groups}}
+|}