Center is quotient-powering-invariant

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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-powering-invariant subgroup)}
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Statement

Suppose G is a group and Z(G) denotes the center of G. Then, Z(G) is a quotient-powering-invariant subgroup of G. Explicitly, if p is a prime number such that G is powered over p, the quotient group G/Z(G) (which can be identified with the inner automorphism group of G) is also powered over p.

Facts used

We essentially use that the center is a fixed-point subgroup of a subgroup of the automorphism group (for Fact (1)) and that it is a central subgroup (for Fact (2)) to get the result.

  1. Center is local powering-invariant (and hence, the center is powering-invariant). This is a special case of the fact that fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
  2. Powering-invariant and central implies quotient-powering-invariant

Proof

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