Finite group in which all partial sum values of squares of degrees of irreducible representations divide the order of the group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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A finite group in which all cumulative conjugacy class size statistics values divide the order of the group is a finite group with the following property: for every natural number , the (sum of squares of all degrees of irreducible representations of for which the degree divides ) divides the order of .
To evaluate this property for a group, it suffices to know the degrees of irreducible representations of . Note that we always consider the degrees of irreducible representations over a splitting field such as or .
In addition to all finite abelian groups, all groups of order , , and have this property for any prime number . Further information: All partial sum values of squares of degrees of irreducible representations divide the order of the group for groups up to prime-fifth order
However, there exist counterexamples of order for every prime number , so the property is not held by every finite nilpotent group or even by every group of prime power order. Further information: There exist groups of prime-sixth order in which the partial sum values of squares of degrees of irreducible representations do not divide the order of the group
The smallest counterexample is symmetric group:S4, which is a group of order 24 where the sum of squares of degrees of irreducible representations dividing 3 is 20.