On the covering number of loops

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G and is denoted by σ(G). It is an interesting problem in group theory to determine integers n such that there exists a group G with σ(G)=n and n−1>1 is not a power of a prime. It is known that there exist integers n>2, which are not covering numbers of a group. So far it has been shown that for n<27 the integers 7, 11, 19, 21, 22 and 25 are not covering numbers of a group. It is an open question if the set of integers with this property is finite or infinite. In this paper an idempotent quasigroup of order n>2 with the property that every two distinct elements generate the entire quasigroup is obtained. With this it is shown that for any integer n>2 there exists a loop whose covering number is n.