Characteristic direct factor of nilpotent group

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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:

  1. 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).
  2. 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.