Another measuring argument for finite permutation groups.

Let G denote a finite group of permutations of a finite set . Given an orbit function : and a class function : G and their extensions to subsets of and G, consider the quantity m,, which is the maximal value of ()( CG()) over all nontrivial subsets of . Denote by the set of non-empty subsets of which satisfy m()( CG() = m,. Our main result, Theorem 1, states that if contains a unique maximal or minimal element with respect to inclusion, then g = for each g G and CG() is a normal subgroup of G. This result may have potential future applications. As an example of an application of Theorem 1 we prove that if either G is a simple group or it is transitive on , T is a normal subset of G not containing 1, is a G-invariant subset of and , then (see Theorem 6). [ABSTRACT FROM AUTHOR]