Powering-injectivity is inherited by extensions where the normal subgroup is contained in the hypercenter

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Suppose we have the following:

such that the following are satisfied:

Then, G is also a \pi-powering-injective group.

Facts used

  1. Powering-injectivity is inherited by central extensions
  2. Something to show that each of the groups (H \cap Z^{i+1}(G))/(H \cap Z^i(G)) is \pi-torsion-free, similar to equivalence of definitions of nilpotent group that is torsion-free for a set of primes.