The automorphism tower of the Mennicke group \(M(-1,-1,-1)\) (Q6569344)

Suppose that \(G\) is a centerless group, i.e., \(G\) has a trivial center. Clearly, \(G\) is isomorphic to the inner automorphism group of \(G\) which is a normal subgroup of the automorphism group of \(G\) and the automorphism group of \(G\) also has trivial center. This process can then be iterated and by making suitable identifications this allows us to construct the automorphism tower of the group \(G\) namely \(G=G_0\triangleleft G_1\triangleleft G_2\triangleleft\dots \triangleleft G_{\alpha}\triangleleft G_{\alpha +1}\triangleleft \cdots\) such that \(C_{G_{\alpha +1}}(G_{\alpha})=1\), \(G_{\alpha +1}\) is the automorphism group of \(G_{\alpha}\) and \(G_{\rho}=\bigcup_{\alpha <\rho}G_{\alpha}\) for all limit ordinals \(\rho\). A theorem of \textit{H. Wielandt} [Math. Z. 45, 209--244 (1939; JFM 65.0061.02)] shows that the automorphism tower of a finite group terminates in finitely many steps. \textit{A. Rae} and \textit{J. E. Roseblade} [Math. Z. 117, 70--75 (1970; Zbl 0213.30101)] extended this to Chernikov groups, whereas \textit{J. A. Hulse} [J. Algebra 16, 347--398 (1970; Zbl 0207.03404)] proved that the automorphism tower of a polycyclic group terminates in at most countably many steps. \textit{S. Thomas} [Proc. Am. Math. Soc. 95, 166--168 (1985; Zbl 0575.20030)] proved that the automorphism tower of any centerless group \(G\) eventually terminates in some ordinal and \textit{J. D. Hamkins} [Proc. Am. Math. Soc. 126, No. 11, 3223--3226 (1998; Zbl 0904.20027)] obtained the same result for all groups, not just centerless ones. Hulse [loc. cit.] also showed that the automorphism tower of the infinite dihedral group terminates in \(\omega +1 \) steps. Although the proofs of Thomas [loc. cit. and Hamkins [loc. cit. mentioned above are relatively short, constructing the automorphism towers of individual groups can often be a long and difficult process. \NIn this paper, the author considers the infinite centerless group \[M=\langle x,y,z\mid x^y=x^{-1}, y^z=y^{-1}, z^x=z^{-1}\rangle\] and determines its automorphism tower, together with the automorphism tower of a characteristic subgroup of \(M\) denoted by \(V\). In the case of \(M\), it turns out that the automorphism tower has length \(2\) whereas the automorphism tower of \(V\) has length \(1\). Thus \(\Aut(\Aut(M)\)) is complete as is \(\Aut(V)\). The paper has many interesting calculations and is worth reading for those working in the area.