Center-fixing implies central factor-extensible

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This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., center-fixing automorphism) must also satisfy the second automorphism property (i.e., central factor-extensible automorphism)
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Statement

Suppose G is a group and \sigma is an automorphism of G such that \sigma is center-fixing: \sigma(z) = z for all z in the center of G. Then, \sigma is a central factor-extensible automorphism of G. In other words, given any group K containing G as a central factor, \sigma can be extended to an automorphism of K.

Note that this in particular shows that if every automorphism of G is center-fixing (e.g., if G is centerless, or has a center of order two), then G is an AEP-subgroup inside any group in which it is a central factor.

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