Connected Abelian Groups in Compact Loops

We shall prove the following theorem concerning compact connected abelian groups: For any element x in a compact connected abelian group there is an element y such that the closure of the cyclic group generated by y is connected and contains x. This is a generalisation of the well known fact that any compact connected group whose topology admits a basis of at most continuum cardinality is monothetic (i.e. contains a dense cyclic subgroup) [ll]. Our proof leans heavily on the theory of duality for locally compact abelian groups which seems to be an appropriate procedure in view of the fact that the character of this theorem is typically abelian; it must remain undecided whether there is an approach using the fact that a compact connected abelian group can be approximated by torus groups. We shall use the theorem mentioned above to determine the structure of compact loops in which every pair of elements generates an abelian subgroup. These loops are called di-associative and commutative. In a compact group the connected component of the identity is the set of all divisible elements [10]; we have been unable to establish the corresponding result for compact di-associative and commutative loops; it is easy to see that divisibility implies connectedness; as long as the converse is not established, the concept of divisibility has to replace the notion of connectivity wherever it occurs in the theory of abelian groups to obtain the analogous results for compact diassociative commutative loops. Our main result is the following: The set D of all divisible elements in a compact di-associative commutative loop G is a closed subgroup of the center of G; no element in G/D other than the identity is divisible and G/D contains no nontrivial connected subgroup. This implies in particular that every compact di-associative commutative divisible loop is a group. Moreover, G/D is a direct product of closed characteristic subloops, one for each prime p, having the property to be divisible by all natural numbers relatively prime to p. Up to the structure of these constituents the structure of compact di-associative and commutative loops is now rather completely described. One would, however, like to know whether the connected component C of the unit in G can actually be larger than the set D of divisible elements.