# Wielandt automorphism tower theorem

I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939).

The theorem states the following: Given a finite, centerless group $G$, its automorphism tower terminates (stabilizes) in finitely many steps.

Proofs of this theorem are available in textbooks. See e.g. a list in Proc. AMS 89(1983), pp 182-183 note by Martin Pettet.

• As far as I can tell, the textbooks referenced discuss bounds on how many steps it takes to stabilize (which is also the topic of the note), but I don't know if they actually cover the proof of Wielandt's theorem (it's probable that they do however). – Justin Benfield Apr 30 '17 at 21:33
• Robinson's book does give a complete proof in Sect. 13.5, and not too long either. And no, it does not discuss explicit bounds, as far as I can tell. – Dima Pasechnik May 1 '17 at 12:52
• It appears to be possible to download a pdf of Robinson's book from ia600204.us.archive.org/16/items/… (I got this link at archive.org/details/springer_10.1007-978-1-4419-8594-1) – Dima Pasechnik May 1 '17 at 12:54

The theorem appears as Theorem 9.10 in

Isaacs, I.Martin, Finite group theory. Graduate Studies in Mathematics 92. Providence, RI: American Mathematical Society (ISBN 978-0-8218-4344-4/hbk). xi, 350~p. (2008). DOI 10.1090/gsm/092. ZBL1169.20001, MR2426855.

The proof given there is very detailed (and well written). (But I think this book is not freely available.)