Moving up and down in the generic multiverse

We give a brief account of the modal logic of the generic multiverse, which is a bimodal logic with operators corresponding to the relations "is a forcing extension of" and "is a ground model of". The fragment of the first relation is called the modal logic of forcing and was studied by us in earlier work. The fragment of the second relation is called the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.Comment: 10 pages. Extended abstract. Questions and commentary concerning this article can be made at http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse

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