Upper central series members are quotient-local powering-invariant in nilpotent group

Statement

Suppose $G$ is a nilpotent group. Then, all members of the upper central series of $G$ are quotient-local powering-invariant subgroups and hence also local powering-invariant subgroups in $G$.

Related facts

Similar facts

• Center is local powering-invariant
• Center is quotient-local powering-invariant in nilpotent group
• Upper central series members are local powering-invariant in Lie ring

Opposite facts

• Second center not is local powering-invariant in solvable group
• Center not is quotient-local powering-invariant in solvable group

Facts used

Style (A)

1. Center is quotient-local powering-invariant in nilpotent group
2. Nilpotency is quotient-closed
3. Quotient-local powering-invariance is quotient-transitive
4. Quotient-local powering-invariant implies local powering-invariant

Style (B)

1. Center is local powering-invariant
2. Local powering-invariance is quotient-transitive in nilpotent group
3. Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group

Proof

The proof method used in this article is discussed in the survey article inductive proof methods for the ascending series corresponding to a subgroup-defining function.|See a list of facts whose proof uses this method

Using Style (A)

The proof of quotient-local powering-invariance follows directly from Facts (1)-(3), using the principle of mathematical induction. The proof of local powering-invariance follows by combining with Fact (4).

Using Style (B)

This proof is also fairly straightforward from the given facts.