### The height of the automorphism tower of a group

by Shelah. [Sh:810]

For a group G with trivial center there is a natural
embedding of G into its automorphism group, so we can look at the
latter as an extension of the group. So an increasing continuous
sequence of groups, the automorphism tower, is defined, the height
is the ordinal where this becomes fixed, arriving to a complete
group. We first show that for any kappa = kappa^{aleph_0}
there is a group of cardinality kappa with height > kappa^+
(improving the lower bound). Second we show that for many such
kappa there is a group of height > 2^kappa, so proving that
the upper bound essentially cannot be improved.

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