Throughout this note, all groups are finite. The study of the sizes of conjugacy classes of groups goes back to the beginning of character theory: W. Burnside proved that a simple group does not have a conjugacy class of size that is a prime power (larger than 1), and this is a key element in his proof of his pa⁢qb-Theorem (see [8, Theorem (3.9)]). N. Itô began the study of the structure of a group in terms of the number of conjugacy class sizes in [10], where he proved that a group having only two conjugacy class sizes must be the direct product of a p-group and an abelian group. K. Ishikawa had a major breakthrough in [12], where he proved that a p-group with only two conjugacy class sizes has nilpotence class at most 3; furthermore, he shows in [11] that a p-group has conjugacy class sizes {1,p} if and only if the group is isoclinic to an extra-special p-group. There has followed a great deal of work on this topic. Rather than trying to summarize this work, we refer the reader to the excellent expository paper by A. R. Camina and R. D. Camina [3].

The study of group elements that are zeros for some irreducible character also goes back to the beginnings of character theory. Burnside also proved that if χ is a nonlinear irreducible character of a group G, then there is some element g∈G so that χ⁢(g)=0 (see [8, Theorem (3.15)]). With this in mind, an element g∈G is said to be vanishing if there exists an irreducible character χ∈Irr⁢(G) so that χ⁢(g)=0. Notice that if one element of a conjugacy class is vanishing, then all elements of that class are vanishing. Hence, we say that a conjugacy class is vanishing if the elements in the conjugacy class are vanishing.

Recently, there has been more concentrated study on the vanishing elements and classes in a group. This began with the work of I. M. Isaacs, G. Navarro and T. R. Wolf in [9], where they proved that if G is solvable, P is a Sylow 2-subgroup of G, and F⁢(G) denotes the Fitting subgroup of G, then every element outside the conjugates of P⁢F⁢(G) is a vanishing element. Furthermore, if G is nilpotent, then they prove that every element outside Z⁢(G) is vanishing, and if G is supersolvable, then every element outside Z⁢(F⁢(G)) is vanishing. In [7, 6], the authors attach a graph to orders of the vanishing elements, and they obtain results regarding which graphs can arise. We suggest the reader consult [5] for a nice expository paper that presents most of the known results regarding vanishing elements.

Recently, there has begun to be research that studies the sizes of the vanishing classes; although, looking at [5], one will see that there are very few results in this subject so far. However, in [4], the third author with S. Dolfi and L. Sanus prove that if a prime p does not divide the size of any vanishing conjugacy class of G, then G has a normal p-complement and abelian Sylow p-subgroups. Also, in [1], vanishing class sizes in nonsolvable groups are studied.

In this paper, we follow the lead of Ishikawa, and we classify the groups where the only vanishing class size is a prime p. The main result of our paper is the following.

Note that, in case (2) of Theorem 1, a Frobenius complement of G/Z⁢(G) is cyclic of order dividing p-1.

We believe there is some hope that Theorem 1 can be extended to groups whose only vanishing class size is a prime power pn. In particular, we would not be surprised if it were true that, under this hypothesis, G is either the direct product of a p-group with class sizes 1 or pn and an abelian p′-group, or G/Z⁢(G) is a Frobenius group whose Frobenius kernel has order pn and whose Frobenius complements are abelian.

However, the converse of this is not true. As an example, let G=SL⁢(2,3): then G/Z⁢(G) is a Frobenius group whose Frobenius kernel has order 4, but the vanishing class sizes of G are 4 and 6. Restricting our attention to Frobenius groups, it is not difficult to see that any Frobenius group where all the vanishing elements lie outside the Frobenius kernel and a Frobenius complement is cyclic will have only one vanishing class size, which is the order of the Frobenius kernel. We do not know of any examples of Frobenius groups whose Frobenius kernel is a p-group for some prime p and a Frobenius complement is cyclic, where there are vanishing elements in the Frobenius kernel (by [9, Theorem A], there are no such examples if the Frobenius kernel is abelian). We believe that proving the existence or nonexistence of such groups will likely involve results similar to those in [2, 13].

We already mentioned the result by Itô which establishes that if a group G has only two conjugacy class sizes, then the unique size larger than 1 is a prime power; as a concluding remark, we note that this does not translate to vanishing class sizes. In fact, it is not difficult to find examples of Frobenius groups, where the Frobenius kernel is not of prime-power order, the Frobenius complements are cyclic, and all the vanishing elements lie outside of the Frobenius kernel: these give examples of groups with only one vanishing class size which is not a prime power.