Stable Extensions of Complete Groups

Abstract: A group is said to be stable if it is isomorphic to its automorphism group. Centerless groups are naturally embedded in their automorphism groups via the map sending an element to conjugation by that element, partially constraining the structure of their automorphisms. As such, it is natural to ask if we can use centerless groups to construct stable groups with nontrivial centers. To this end, we classify all finite stable groups arising as central extensions of centerless groups. Furthermore, all finite stable groups arising as extensions of centerless groups by groups of nilpotency class two with trivial induced outer action on the kernel are classified. Finally, it is shown that there are infinitely many stable groups of each of the above two types. As a corollary we show that there are infinitely many non-stable finite groups equinumerous with their automorphism groups.