Stable Extensions of Complete Groups

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.

💡 Research Summary

The paper investigates “stable” finite groups—those that are isomorphic to their own automorphism groups—by exploiting the well‑understood structure of complete (centerless, outer‑automorphism‑free) groups. The author’s central strategy is to start with a center‑less complete group (K) and to form extensions of two restricted types: (1) central extensions (1\to N\to G\to K\to1) where the kernel (N) lies in the centre of (G), and (2) extensions by a nilpotent group of class two (N) whose induced outer action on the kernel is trivial.

The first technical block (Lemmas 3.1–3.3) shows that if the outer action is trivial, the nilpotent kernel (N) and the centraliser (C_G(N)) together generate the whole group, and the upper central series of (N) coincides with that of (G). Consequently both (N) and (C_G(N)) are characteristic in (G) and the automorphism group splits as
\