A note on groups with just-infinite automorphism groups. (Q2856422)

A group \(G\) is just-infinite if and only if it is infinite but all of its proper homomorphic images are finite.NEWLINENEWLINE The authors prove the following theorem. Theorem: Let \(G\) be a group admitting an ascending normal series whose factors are either central or finite. Then the automorphism group \(\Aut(G)\) is not just-infinite.NEWLINENEWLINE In particular, hypercentral groups cannot have just-infinite automorphism groups and the upper central series of any group with a just-infinite automorphism group terminates at the center. Finally, if \(G\) is an infinite simple group whose outer automorphism group \(\mathrm{Out}(G)\) is finite, then \(\Aut(G)\) is just-infinite.