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 paβ’qb-Theorem (see [8, Theorem (3.9)]).
N. ItΓ΄ 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.βR. Camina and R.βD. 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 Ο is a nonlinear irreducible character of a group G, then there is some element gβG so that Οβ’(g)=0 (see [8, Theorem (3.15)]).
With this in mind, an element gβG is said to be vanishing if there exists an irreducible character ΟβIrrβ’(G) so that Οβ’(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.βM. Isaacs, G. Navarro and T.βR. Wolf in [9], where they proved that if G is solvable, P is a Sylow 2-subgroup of G, and Fβ’(G) denotes the Fitting subgroup of G, then every element outside the conjugates of Pβ’Fβ’(G) is a vanishing element.
Furthermore, if G is nilpotent, then they prove that every element outside Zβ’(G) is vanishing, and if G is supersolvable, then every element outside Zβ’(Fβ’(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.
Theorem 1.
Let G be a group, and let p be a prime.
The vanishing classes of G all have size p if and only if one of the following occurs:
G=PΓH, where P is a p-group whose conjugacy class sizes are
1
or p, and H is an abelian pβ²-group.
G/Zβ’(G) is a Frobenius group whose Frobenius kernel has order p.
Note that, in case (2) of Theorem 1, a Frobenius complement of G/Zβ’(G) is cyclic of order dividing p-1.
Proof.
Assume first that G is nilpotent: then we claim that all the vanishing conjugacy classes of G have size p if and only if G is as in (1).
In fact, under the nilpotency assumption, [9, Theorem B] shows that the vanishing elements of G are precisely the elements in GβZβ’(G); hence, the set of vanishing conjugacy class sizes of G is actually the whole set of conjugacy class sizes of G, excluding 1.
As is easily seen, the conjugacy class sizes of G are all 1 or p if and only if G has the structure described in (1), so the theorem is proved in this situation.
In view of the above discussion, we will henceforth assume that G is not nilpotent.
As the next step we show that if the structure of G is as in (2), then the vanishing conjugacy class sizes of G are all p.
So, let G/Zβ’(G) be a Frobenius group whose Frobenius kernel has prime order p, and let H,K be subgroups of G such that K/Zβ’(G) and H/Zβ’(G) are the Frobenius kernel and a Frobenius complement of G/Zβ’(G), respectively.
Note that, as both the Sylow p-subgroup K/Zβ’(G) and the Hall pβ²-subgroup H/Zβ’(G) of G/Zβ’(G) are cyclic, every Sylow subgroup of G/Zβ’(G) is cyclic.
In particular, every irreducible character of Zβ’(G) has an extension to G (see [8, Theorem (6.26) and Corollary (11.22)]).
Now, let Ο be in Irrβ’(G), and let ΞΌ be the unique irreducible constituent of ΟZβ’(G); since ΞΌ has an extension Ξ·βIrrβ’(G), by Gallagherβs Theorem ([8, Corollary (6.17)]), there exists ΟβIrrβ’(G) such that Ο=Ξ·β’Ο and kerβ’(Ο)βZβ’(G).
As Ξ· is linear, for any xβG, we have Οβ’(x)=0 if and only if Οβ’(x)=0; from this, it is easily seen that x is a vanishing element of G if and only if xβ’Zβ’(G) is a vanishing element of G/Zβ’(G).
Finally, an application of [9, Theorem A] yields that every element of K/Zβ’(G) is non-vanishing in G/Zβ’(G); as a consequence, if x is a vanishing element of G, then xβ’Zβ’(G) lies in a Frobenius complement of G/Zβ’(G) (i.e., x lies in a conjugate of H) and, as H is abelian, the class size of x is p, as wanted.
It remains to prove that if G is not nilpotent and every vanishing conjugacy class of G has size p, then G/Zβ’(G) is a Frobenius group whose Frobenius kernel has order p.
By [4, Theorem A], under our assumptions, G has a normal q-complement and abelian Sylow q-subgroups for every prime q other than p.
Therefore, G has a normal Sylow p-subgroup P, and a Hall pβ²-subgroup H of G is nilpotent because it has a normal q-complement for every prime q dividing its order, but H is in fact abelian as all its Sylow subgroups are abelian.
Observe also that, since H is abelian and Irrβ’(P/Ξ¦β’(P)) can be viewed as a completely reducible H-module, there exists Ξ»βIrrβ’(P) (such that kerβ’(Ξ»)βΞ¦β’(P)) whose inertia subgroup in G is Pβ’CHβ’(P).
So, if Ξ²βIrrβ’(Pβ’CHβ’(P)) is such that Ξ» is a constituent of Ξ²P, Clifford Correspondence yields that Ξ²G is in Irrβ’(G); moreover, since Pβ’CHβ’(P) is a normal subgroup of G, Ξ²G vanishes outside Pβ’CHβ’(P), and therefore every element in GβPβ’CHβ’(P) is vanishing in G.
By our assumptions, we get |G:CG(x)|=p for every xβGβPβ’CHβ’(P).
As G is not nilpotent, we have CHβ’(P)β H, and therefore we can choose h in HβCHβ’(P).
Now, assume that yβP commutes with h.
Since CHβ’(P) is the unique Hall pβ²-subgroup of Pβ’CHβ’(P), our choice of h ensures that both h and hy do not lie in Pβ’CHβ’(P); hence we get
|G:CG(h)|=p=|G:CG(hy)|
by the paragraph above.
But, as h and y are commuting elements of coprime orders, we get CGβ’(hβ’y)=CGβ’(h)β©CGβ’(y), whence CGβ’(h)=CGβ’(hβ’y).
Moreover, as H is abelian, we have Hβ€CGβ’(h)=CGβ’(hβ’y)β€CGβ’(y).
We conclude that CPβ’(H)=CPβ’(h) has index p in P, thus CPβ’(H)β’β΄β’G.
Next, we work to show that CPβ’(H) is central in G.
Let z be in CPβ’(H), and choose h in HβCHβ’(P):
by the paragraph above, we have CGβ’(h)β€CGβ’(z).
Thus, we deduce that CPβ’(H)=CPβ’(h) centralizes z; as this holds for every zβCPβ’(H), we see that CPβ’(H) is abelian.
In particular, the centralizer in G of CPβ’(H) contains the subgroup CPβ’(H)β’H, whose index in G equals |P:CP(H)|=p; on the other hand, CGβ’(CPβ’(H)) is normal in G, and if it coincides with CPβ’(H)β’H (which clearly has H as a characteristic subgroup), then H is normal in G, a contradiction.
As a consequence, the whole of G centralizes CPβ’(H).
We are now in a position to finish the proof.
Set Z=CPβ’(H)β’CHβ’(P); in view of the above paragraph (and the fact that H is abelian), we have Zβ€Zβ’(G).
As Pβ’Z/Z is a normal subgroup of G/Z having order p, and which is acted on faithfully by Hβ’Z/Z, we have that G/Z=Pβ’H/Z is a Frobenius group whose Frobenius kernel is Pβ’Z/Z, and Z=Zβ’(G).
Conclusion (2) holds, and our proof is complete.
β
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 pβ²-group, or G/Zβ’(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=SLβ’(2,3):
then G/Zβ’(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 ItΓ΄ 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.
Acknowledgements
Much of the work of this paper occurred during a visit of the 2nd author to UniversitΓ degli Studi di Milano.
He would like to thank his hosts for their hospitality.
The authors would like to thank Marcel Herzog for a number of helpful conversations about this work, and also the referee for her/his valuable comments, which considerably improved and shortened some arguments in the proof of the main theorem.
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