The group of squarefree integers

We investigate the consequences of the elementary observation that the squarefree numbers form a group under the operation lcm/gcd. In particular, we discuss the characters on this group, one of which is the Mobius function, as well as the finite subgroups 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 N. (C) 2014 Elsevier Inc. All rights reserved.