Characteristic direct factor of nilpotent group: Difference between revisions

Latest revision as of 20:15, 14 July 2013

This article describes a property that arises as the conjunction of a subgroup property: characteristic direct factor with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A subgroup $H$ of a group $G$ is termed a characteristic direct factor of nilpotent group if it satisfies the following equivalent conditions:

$G$ is a nilpotent group and $H$ is a characteristic direct factor of $G$ (i.e., $H$ is both a characteristic subgroup of $G$ and a direct factor of $G$ ).
$G$ is a nilpotent group and $H$ is a fully invariant direct factor of $G$ (i.e., $H$ is both a fully invariant subgroup of $G$ and a direct factor of $G$ ). This has other equivalent formulations; see equivalence of definitions of fully invariant direct factor.

Equivalence of definitions

Further information: equivalence of definitions of characteristic direct factor of nilpotent group

The equivalence follows indirectly from the fact that nontrivial subgroup of nilpotent group has nontrivial homomorphism to center.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic direct factor of abelian group				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant direct factor				\|FULL LIST, MORE INFO
characteristic direct factor				\|FULL LIST, MORE INFO
fully invariant subgroup of nilpotent group				\|FULL LIST, MORE INFO
characteristic subgroup of nilpotent group				\|FULL LIST, MORE INFO
direct factor of nilpotent group				\|FULL LIST, MORE INFO
fully invariant subgroup				\|FULL LIST, MORE INFO
characteristic subgroup				\|FULL LIST, MORE INFO
direct factor				\|FULL LIST, MORE INFO

@@ Line 13: / Line 13: @@
 The equivalence follows indirectly from the fact that [[nontrivial subgroup of nilpotent group has nontrivial homomorphism to center]].
+==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::characteristic direct factor of abelian group]] || || || || {{intermediate notions short|characteristic direct factor of nilpotent group|characteristic direct factor of abelian group}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::fully invariant direct factor]] || || || || {{intermediate notions short|fully invariant direct factor|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::characteristic direct factor]] || || || || {{intermediate notions short|characteristic direct factor|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::fully invariant subgroup of nilpotent group]] || || || || {{intermediate notions short|fully invariant subgroup of nilpotent group|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::characteristic subgroup of nilpotent group]] || || || || {{intermediate notions short|characteristic subgroup of nilpotent group|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::direct factor of nilpotent group]] || || || || {{intermediate notions short|direct factor of nilpotent group|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::fully invariant subgroup]] || || || || {{intermediate notions short|fully invariant subgroup|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::characteristic subgroup]] || || || || {{intermediate notions short|characteristic subgroup|characteristic direct factor of nilpotent group}}
+|-
+| [[Stronger than::direct factor]] || || || || {{intermediate notions short|direct factor|characteristic direct factor of nilpotent group}}
+|}