Abelian profinite groups and the discontinuous isomorphism problem

We investigate the question: ``Can there be a non-continuous isomorphism between two profinite groups which are not topologically isomorphic?" On one end of the spectrum, we show that branch and semisimple profinite groups have no non-continuous automorphisms. On the other, many abelian pro-$p$ groups are abstractly but not topologically isomorphic. The question for countably-based profinite groups was totally answered in a previous publication. There are many examples of such groups which are abstractly but not topologically isomorphic: we give explicit constructions of such non-topological isomorphisms We used Pontryagian duality to reduce the question of classifying countably based abelian pro-$p$ groups to that of countable abelian $p$-groups. In the 1930s Ulm and Zippin classified countable abelian $p$-groups. This work was expanded in the 1970s, to give the theory of totally projective abelian $p$-groups. We survey the structural theory of these groups and construct their duals, the totally injective groups. These provide more positive answers to our question: every dual-reduced totally injective pro-$p$ group is abstractly isomorphic to the closure of its torsion subgroup, although in most cases these groups are not topologically isomorphic. We proceed to give a detailed discussion on the features of the abstract and of the topological subgroup structures of such groups. We introduce a new invariant, unbounded multiplicity, of Cartesian products of finite $p$-groups, in the above proof. This allows us to use infinite combinatorial arguments which give more results. Two of these Cartesian groups are isomorphic modulo their torsion subgroups if and only if they have the same unbounded multiplicity. A totally injective pro-$p$ group will be abstractly isomorphic to its closed torsion subgroup whenever the unbounded multiplicity of this subgroup bounds the dimension of continuous torsion-free quotients. Additionally, we construct a new class of commutative, unital pro-$p$ rings. For each totally injective abelian pro-$p$ group $G$, we construct a pro-$p$ ring $R$ with $(R,+)=G$.Open Acces