Let G be a group. G is called an n-abelian group if \((xy)^ n=x^ ny^ n\) for the integer n and all x,y\(\in G\). G is called an n-Levi group if \((x^ n,y)=(x,y)^ n\) for the integer n and all x,y\(\in G\). G is called an n-Bell group if \((x^ n,y)=(x,y^ n)\) for the integer n and all x,y\(\in G.\) The exponent semigroup of G is defined as the set of all integers n so that G is n-abelian. The Levi semigroup of G is defined as the set of all integers n so that G is n-Levi. The Bell semigroup of G is defined as the set of all integers n so that G is n-Bell. The author obtains arithmetic characterizations of the exponent semigroup of G, the Levi semigroup of G, and the Bell semigroup of G. She finds in Theorem 1, her main result, that these characterizations are the same in all three cases. She concludes by deriving consequences of Theorem 1 and exhibiting interesting examples.