The cofinality spectrum of the infinite symmetric group
  A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of lambda proper subgroups. The cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove   Theorem (A): Let T be any subset of omega setminus {0}. Then it is consistent that aleph_n in CF(S) if and only if n in T .   One might suspect that it is consistent that CF(S) is an arbitrarily prescribed set of regular uncountable cardinals, subject only to the above mentioned constraint. This is not the case.   Theorem (B): If aleph_n in CF(S) for all n in omega setminus {0}, then aleph_{omega +1} in CF(S) . 
Saharon Shelah, Simon Thomas