For orders and conjugacy in finite group theory, Lagrange?s Theorem and the class equation have universal applications. Recently, the last two authors obtained order and conjugacy class formulas for factorizable monoids, M(G) induced by certain subgroups G of the symmetric group S_n. In particular, when G is balanced, that is, when subsets of {1, 2, ?, n} of the same size have G-stabilizers of the same order, a formula was derived for the order of M(G). In this article, we investigate balanced groups as a class of permutation groups. We obtain a characterization of these groups in terms of the properties of multiple transitivity and semiregularity of subgroups of S_n. In addition, we exhibit several infinite classes of balanced groups using permutations of finite fields and classify all balanced subgroups of S_n for n ? 15.

balanced groups, permutation groups, multiply transitive groups, semiregular groups, factorizable monoids, class equation.