# 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) neednotsatisfy 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.