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

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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

Opposite facts

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.