Local powering-invariance is quotient-transitive in nilpotent group

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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 G is a nilpotent group with subgroups H \le K satisfying the following:

Then, K is a local powering-invariant subgroup of G.

Related facts

Applications

Opposite facts

Facts used

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

Proof

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