Group elements of prime power index

Abstract

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of prime power index, not 1, in simple groups. From Burnside's Theorem one deduces easily that a group without proper characteristic subgroups contains an element, not 1, whose index is a power of a prime if and only if this group is abelian. In this result it suffices to assume the absence of proper fully invariant subgroups, since we can prove [in ?2] the rather surprising result that a [finite] group does not possess proper fully invariant subgroups if and only if it does not possess proper characteristic subgroups. A deeper insight will be gained if we consider groups which contain many elements of prime power index. We show [in ?5 ] that the elements of order a power of p form a direct factor if, and only if, their indices are powers of p too; and nilpotency is naturally equivalent to the requirement that this property holds for every prime p. More difficult is the determination of groups with the property that every element of prime power order has also prime power index [?3]. It follows from Burnside's Theorem that such groups are soluble; and it is clear that a group has this property if it is the direct product of groups of relatively prime orders which are either p-groups or else have orders divisible by only two different primes and furthermore have abelian Sylow subgroups. But we are able to show conversely that every group with the property under consideration may be represented in the fashion indicated. In ?5 we study the so-called hypercenter. This characteristic subgroup has been defined in various ways: as the terminal member of the ascending central chain or as the smallest normal subgroup modulo which the center is 1. We may add here such further characterizations as the intersection of all the normalizers of all the Sylow subgroups or as the intersection of all the maximal nilpotent subgroups; and the connection with the index problem is obtained by showing that a normal subgroup is part of the hypercenter if, and only if, its elements of order a power of p have also index a powrer of p. Notation. All the groups under consideration will be finite. An element [group] is termed primary, if its order is a prime power;