104 research outputs found
Every group has a terminating transfinite automorphism tower
The automorphism tower of a group is obtained by computing its automorphism
group, the automorphism group of THAT group, and so on, iterating
transfinitely. Each group maps canonically into the next using inner
automorphisms, and so at limit stages one can take a direct limit and continue
the iteration. The tower is said to terminate if a fixed point is reached, that
is, if a group is reached which is isomorphic to its automorphism group by the
natural map. This occurs if a complete group is reached, one which is
centerless and has only inner automorphisms. Wielandt [1939] proved the
classical result that the automorphism tower of any centerless finite group
terminates in finitely many steps. Rae and Roseblade [1970] proved that the
automorphism tower of any centerless Cernikov group terminates in finitely many
steps. Hulse [1970] proved that the the automorphism tower of any centerless
polycyclic group terminates in countably many steps. Simon Thomas [1985] proved
that the automorphism tower of any centerless group eventually terminates. In
this paper, I remove the centerless assumption, and prove that every group has
a terminating transfinite automorphism tower.Comment: 4 pages, to appear in the Proceedings of the American Mathematical
Society, see also
http://scholar.library.csi.cuny.edu/users/hamkins/papers.html#MyAutoTower
On spectral disjointness of powers for rank-one transformations and M\"obius orthogonality
We study the spectral disjointness of the powers of a rank-one
transformation. For a large class of rank-one constructions, including those
for which the cutting and stacking parameters are bounded, and other examples
such as rigid generalized Chacon's maps and Katok's map, we prove that
different positive powers of the transformation are pairwise spectrally
disjoint on the continuous part of the spectrum. Our proof involves the
existence, in the weak closure of {U_T^k: k in Z}, of "sufficiently many"
analytic functions of the operator U_T. Then we apply these disjointness
results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic)
symbolic models associated to these rank-one constructions: All sequences
realized in these models are orthogonal to the M\"obius function
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