The positive integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once.

The positive integer n is square-free if and only if μ ( n ) ≠ 0, where μ denotes the Möbius function.

The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic.

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