Center is quotient-local powering-invariant in nilpotent group

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This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., quotient-local powering-invariant subgroup)}
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Statement

Suppose G is a nilpotent group and Z(G) is the center of G. Then, Z(G) is a quotient-local powering-invariant subgroup of G: if any element of G has a unique p^{th} root in G for some prime number p, then its image in G/Z(G) has a unique p^{th} root in G/Z(G).

Related facts

Opposite facts

Facts used

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

Proof

The proof follows directly by combining Facts (1), (2), and (3).