Characteristic direct factor of nilpotent group
From Groupprops
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
Contents
Definition
A subgroup 
of a group 
is termed a characteristic direct factor of nilpotent group if it satisfies the following equivalent conditions:
- is a nilpotent group and is a characteristic direct factor of (i.e., is both a characteristic subgroup of and a direct factor of ).
- is a nilpotent group and is a fully invariant direct factor of (i.e., is both a fully invariant subgroup of and a direct factor of ). 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 |
