A note on finite groups with few values in a column of the character table

REND. SEM. MAT. UNIV. PADOVA, Vol. 115 (2006)

A Note on Finite Groups with Few Values in a Column of the Character Table.

MARIAGRAZIABIANCHI(*) - DAVIDCHILLAG(**) - EMANUELEPACIFICI(***) Dedicated to Professor Guido Zappa on the occasion of his 90th birthday

ABSTRACT- Many structural properties of a finite groupGare encoded in the set of irreducible character degrees ofG. This is the set of (distinct) values appearing in the ``first'' column of the character table ofG. In the current article, we study groups whose character table has a ``non-first'' column satisfying one particular condition. Namely, we describe groups having a nonidentity element on which all nonlinear irreducible characters take the same value.

As many results in the literature show, the structure of a finite groupG is deeply reflected and influenced by certain arithmetical properties of the set of irreducible character degrees ofG. This is in fact the set of (distinct) values which the irreducible characters ofGtake on the identity element.

Our aim in this note is to describe groups having anonidentityelement on which all nonlinear irreducible characters take the same value, although in the Theorem below we shall consider a situation which is (in principle, and from one point of view) more general. Therefore, our result can be com- pared with studies on groups whose nonlinear irreducible characters have all the same degree (see for instance [3, Chapter 12], and [1]).

(*) Indirizzo dell'A.: Dipartimento di Matematica ``Federigo Enriques'', Uni- versitaÁ degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy.

E-mail: bianchi@mat.unimi.it

(**) Indirizzo dell'A.: Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel.

E-mail: chillag@techunix.technion.ac.il

(***) Indirizzo dell'A.: Dipartimento di Matematica ``Federigo Enriques'', UniversitaÁ degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy.

E-mail: pacifici@mat.unimi.it

In what follows, every group is tacitly assumed to be finite. We shall denote by Lin (G) the set of linear characters of the groupG; also, ifgis an element ofG, the symbolg^Gwill denote the conjugacy class ofg.

THEOREM. Let G be a group with centre Z, and let g be a nonidentity element of the derived subgroup G⁰. Assume that all the values of the nonlinear irreducible characters of G on g are rational and negative. Then g^GˆG⁰n f1g(so that G⁰is an elementary abelian minimal normal sub- group of G, say of order pⁿfor a suitable prime p), and one of the following holds.

(a) G is a direct product of a2-group with an abelian group, and G⁰ˆ hgihas order2(whence it lies in Z).

(b) G⁰\Zˆ1, and G=Z is a Frobenius group with elementary abelian kernel G⁰Z=Z of order pⁿ>2 and cyclic complement of order pⁿ 1.

Conversely, we have the following.

(a⁰) Let G be a group as in(a). Then G has exactlyjZj=2nonlinear irreducible characters, all of the same degree d. Here d²ˆ jG:Zj, so d is a power of 2 different from1. Also, every nonlinear irreducible character of G takes value d on the nonidentity element of G⁰.

(b⁰) Let G be a group as in (b). Then G has exactlyjZjnonlinear irreducible characters, all of the same degree dˆ jG:G⁰Zj ˆpⁿ 1. Also, every nonlinear irreducible character of G takes value 1on every non- identity element of G⁰.

It may be worth stressing that, assuming the hypotheses of the The- orem, the values which the nonlinear irreducible characters ofGtake ong turn out to be the same value v, and we have vˆ 2^s for some s0.

Moreover,Gis of type (a) precisely when s6ˆ0, whereas it is of type (b) precisely whensˆ0.

As an immediate consequence of the Theorem, we obtain the following.

COROLLARY. Let G be a nonabelian group, and g a nonidentity element ofGsuch that every nonlinear irreducible character ofGtakes the same valuevong. Then eithervˆ0, which occurs if and only ifgdoes not lie in the derived subgroupG⁰, org^GˆG⁰n f1g, andGis as in(a)or(b) of the Theorem.

We present next a proof of the main result of this note.

162 Mariagrazia Bianchi - David Chillag - Emanuele Pacifici

PROOF. [Proof of the Theorem] By the second orthogonality relation ([3, 2.18]), we get

0ˆ X

x2Irr (G)

x(g)x(1)ˆ jG:G⁰j ‡ X

x2Irr (G)nLin (G)

x(g)x(1):

Suppose now that there exists y2G⁰n f1g which is not conjugate to g.

Then, as above,

0ˆ X

x2Irr (G)

x(y)x(g)ˆ jG:G⁰j ‡ X

x2Irr (G)nLin (G)

x(y)x(g):

Let us define

G :ˆ X

x2Irr (G)nLin (G)

x(g)x:

As the x(g) are positive integers,Gis a character ofG.

The two equations above implyG(y)ˆG(1). Thusylies inkerG, which is given byT

kerx jx2Irr (G)nLin (G)

f g. On the other handyis inG⁰, so it lies in the kernel of every linear character of Gas well. Since the in- tersection of all the kernels of irreducible characters of a finite group is trivial, this yields a contradiction.

The conclusion so far is that G⁰n f1g is in fact a conjugacy class, and nowG⁰is a minimal normal subgroup ofGin which all nonidentity elements have the same order. Therefore G⁰ is an elementary abelianp-group for some primep, say of orderpⁿ. It is clear that eitherG⁰Z, orG⁰\Zˆ1.

IfG⁰ lies inZ, then Gis nilpotent (of class 2). Also, as jG⁰j is forced to be 2, all the Sylow subgroups of G are abelian except the Sylow 2-subgroup. In other words,Gis as in (a) of the statement. It is well known that a group of this type satisfies the properties listed in (a⁰) (see for ex- ample 7.5 and 7.6(a) in [2]).

In the case G⁰\Zˆ1, we use an argument as in [1, Theorem 1].

Namely, there exist a primeq6ˆpand a Sylowq-subgroup ofG(call itQ) such that Q does not centralize G⁰. In particular Q is not normal, so T:ˆN_G(Q) is a proper subgroup of G. An application of the Frattini argument yields nowGˆTG⁰and, asG⁰is abelian, we see thatT\G⁰is a normal subgroup ofG. By the minimality ofG⁰, and since Tis a proper subgroup of G, we get T\G⁰ˆ1. It follows that T is abelian, whence CT(G⁰)ˆZ. Now, G⁰ can be viewed as a faithful simple module forT=Z over the prime fieldGF(p) (the action being defined by conjugation); the fact that there exists such a module for T=Z implies that T=Z is cyclic.

The order ofT=Zequals the length of an orbit inG⁰n f1g, that ispⁿ 1.

A note on finite groups with few values in a column etc. 163

It is now clear that G is a group of the type described in (b) of the statement.

It remains to show that every group as in (b) satisfies the properties listed in (b⁰) of the statement. Let us consider the abelian normal subgroup A:ˆZG⁰ofG, whose irreducible characters are of the formlmforl in Irr (Z) and m in Irr (G⁰). SinceG acts transitively on the elements of G⁰n f1g, it acts transitively on the set of nonprincipal irreducible char- acters ofG⁰as well (see [2, 18.5(c)]). Thus, fixedm6ˆ1_G⁰ in Irr (G⁰), we get I_G(lm) ˆA for all l in Irr (Z). We conclude that the map l7!x_l:ˆ(lm)"^G, defined on Irr (Z), has image in Irr (G)nLin (G);

taking into account that all thex_lhave the same degreedˆpⁿ 1, it is easy to check that such map is indeed a bijection. This tells us that all the nonlinear irreducible characters of G have the same degree pⁿ 1;

moreover, we getx_l#_G⁰ˆr_G0 1_G⁰ for alllin Irr (Z) (herer_G0 denotes the regular character ofG⁰), whencex_l(g)ˆ 1 for allginG⁰n f1g.

As for the Corollary, let us consider first the case in which the elementg is not inG⁰. Then there exists a linear charactersofGsuch thats(g)6ˆ1.

Now, if x is in Irr (G)nLin (G), our assumption implies (sx)(g)ˆx(g), which forces x(g) to be 0. We shall then assume g2G⁰. The second or- thogonality relation yields

0ˆ X

x2Irr (G)

x(g)x(1)ˆ jG:G⁰j ‡v X

x2Irr (G)nLin (G)

x(1);

sovis a negative rational number (note that this provesx(g)6ˆ0). We are now in a position to apply the Theorem, and the claim follows.

Acknowledgments. The authors wish to thank the Referee for his va- luable advice.

REFERENCES

[1] M. BIANCHI- A. GILLIO- M. HERZOG- G. QIAN- W. SHI,Characterization of non-nilpotent groups with two irreducible character degrees, J. Algebra284 (2005), pp. 326-332.

[2] B. HUPPERT,Character Theory of Finite Groups, De Gruyter, Berlin, 1988.

[3] I.M. ISAACS,Character Theory of Finite Groups, Academic Press, New York, 1976.

Manoscritto pervenuto in redazione il 10 dicembre 2005.

164 Mariagrazia Bianchi - David Chillag - Emanuele Pacifici