Center-fixing implies central factor-extensible
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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Suppose is a group and is an automorphism of such that is center-fixing: for all in the center of . Then, is a central factor-extensible automorphism of . In other words, given any group containing as a central factor, can be extended to an automorphism of .
Note that this in particular shows that if every automorphism of is center-fixing (e.g., if is centerless, or has a center of order two), then is an AEP-subgroup inside any group in which it is a central factor.