The group of squarefree integers

We investigate the consequences of the elementary observation that the squarefree numbers form a group under the operation lcmgcd. In particular, we discuss the characters on this group, one of which is the Möbius function, as well as the finite subgroups D(k)D(k) formed from the divisors of a given squarefree integer k. We show further how a convolution, naturally based on this operation, leads to the factorization of various arithmetical matrices and the evaluation of the eigenvalues. We discuss briefly the associated L  -functions. Finally, we generalize the operation to other subsets of NN.