Characteristic not implies powering-invariant in nilpotent group
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup of nilpotent group) need not satisfy the second subgroup property (i.e., powering-invariant subgroup of nilpotent group)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about characteristic subgroup of nilpotent group|Get more facts about powering-invariant subgroup of nilpotent group
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup of nilpotent group but not powering-invariant subgroup of nilpotent group|View examples of subgroups satisfying property characteristic subgroup of nilpotent group and powering-invariant subgroup of nilpotent group
Statement
It is possible to have a nilpotent group and a characteristic subgroup
of
such that
is not a powering-invariant subgroup of
. In other words, there exists a prime number
such that every element of
has a unique
root, but there are elements of
whose
roots are outside
.
Related facts
Opposite facts
- Characteristic subgroup of abelian group implies powering-invariant
- Upper central series members are powering-invariant
- Lower central series members are powering-invariant in nilpotent group
- Derived series members are powering-invariant in nilpotent group
Similar facts
Proof
See the example in the references.