Characteristic direct factor of abelian 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: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
This article describes a property that arises as the conjunction of a subgroup property: fully invariant direct factor with a group property imposed on the ambient group: abelian 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
if the following equivalent conditions are satisfied:
-
is an abelian group and
is a characteristic direct factor of
(i.e.,
is both a characteristic subgroup of
and a direct factor in
).
-
is an abelian group and
is a fully invariant direct factor of
(i.e.,
is a fully invariant subgroup as well as a direct factor of
). See also equivalence of definitions of fully invariant direct factor for other equivalent formulations of this.
Equivalence of definitions
Further information: equivalence of definitions of characteristic direct factor of abelian group