# Local powering-invariance is quotient-transitive in nilpotent group

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., local powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)

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## Statement

Suppose is a nilpotent group with subgroups satisfying the following:

- is a normal subgroup of .
- is a local powering-invariant subgroup of , i.e., for and such that the equation has a unique solution for in , we must have .
- is a local powering-invariant subgroup of the quotient group .

Then, is a local powering-invariant subgroup of .

## Related facts

### Applications

### Opposite facts

## Facts used

- Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
- Local powering-invariant over quotient-local powering-invariant implies local powering-invariant

## Proof

The proof follows directly by combining Facts (1) and (2).