Powering-invariant normal subgroup of nilpotent group
From Groupprops
This article describes a property that arises as the conjunction of a subgroup property: powering-invariant normal subgroup 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
This article describes a property that arises as the conjunction of a subgroup property: quotient-powering-invariant subgroup 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 of a group
is termed a powering-invariant normal subgroup of nilpotent group if it satisfies the following equivalent conditions:
-
is a nilpotent group and
is a powering-invariant normal subgroup of
, i.e.,
is a normal subgroup and is a powering-invariant subgroup of
. Here, by powering-invariant, we mean if
is a prime number such that
is
-powered,
is also
-powered.
-
is a nilpotent group and
is a quotient-powering-invariant subgroup of
, i.e.,
is a normal subgroup and if
is a prime number such that
is
-powered, the quotient group
is also
-powered.
Equivalence of definitions
- (1) implies (2) follows from normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication.
- (2) implies (1) follows from quotient-powering-invariant implies powering-invariant.