A criterium for monomiality

REND. SEM. MAT. UNIV. PADOVA, Vol. 131 (2014) DOI 10.4171/RSMUP/131-15

A criterium for monomiality

CAROLINAVALLEJORODRIÂGUEZ

ABSTRACT- LetGbe a solvable group. An odd degree rational valued characterx ofGis induced from a linear character of some subgroup ofG. We extend this result to odd degree charactersxofGthat take values in certain cyclotomic extensions ofQ.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 20C15.

KEYWORDS. Monomial characters, characters of finite groups.

1. Introduction

It follows from a well-known result of R. Gow [1] that an odd-degree rational-valued irreducible character of a solvable group is monomial. In this note, we slightly generalize Gow's result.

Ifxis an irreducible complex character of a finite groupG, let n(x)ˆgcd(njQ(x)Q_n);

whereQ(x) is the smallest field containing the values ofx, and we denote by Q_nthen-th cyclotomic field.

THEOREMA. Let G be a solvable group. Letx2Irr…G†. Ifx(1)is odd and gcd(x(1);n(x))ˆ1, then there exist a subgroup UG and a linear characterlof U such thatl^Gˆx. Moreover, ifmis a linear character of some subgroup WG and m^Gˆx, then there exists g2G such that W ˆU^gandmˆl^g.

(*) Indirizzo dell'A.: Departamento de AÂlgebra, Facultad de Ciencias MatemaÂ- ticas C/ Dr. Moliner, 50 C.P. 46100 Burjassot, Spain.

E-mail: carolina.vallejo@uv.es

This research is partially supported by Proyecto MTM2010-15296 of the Spanish Government.

Theorem A is not true ifx(1) is even or ifGis not solvable (SL2(3) has a rational valued non-monomial character of degree 2, andA6has a rational valued non-monomial character of degree 5).

I would like to thank M. Isaacs and G. Navarro for simplifications of earlier versions of the proof of this result.

2. Proof of Theorem A

We use the notation of [2]. First, we prove by induction onjGjthatxis monomial.

STEP1. We may assume xis faithful and that there is no proper sub- groupH<Gandc2Irr…H†such thatc^GˆxandQ(c)Q(x).

LetKˆkerx. IfK>1, all the hypotheses hold inG=Kso by induction we are done. For the second partc(1) dividesx(1), andn(c) dividesn(x), so gcd(c(1);n(c))ˆ1, and the inductive hypothesis applies.

STEP2.F…G† ˆ Q

pj^ÿx(1)O_pG.

Letpbe a prime. Suppose thatpdividesx(1). In particularpis odd, and p does not divide n(x). Let M be a normal p-subgroup of G. Since Q(x)Qn(x)we have thatQ_jMj\Q(x)ˆQ. Hencex_Mis rational-valued. If j2Irr…M†, then [x_M;j]ˆ[x_M;j]. Sincex(1) is odd, there exists a real ir- reducible constituent jof x_M. Since jMjis odd, we have thatjˆ1_M, by Burnside's theorem. By Step 1, we know thatxis faithful and we conclude Mˆ1.

STEP3.FˆF…G†is abelian.

Let Mbe a normal p-subgroup ofG, where pdoes not dividex(1). It then follows that the irreducible constituents of x_M are linear. Let l2Irr…M†be underx. We have thatM⁰kerl^gˆkerl^gfor everyg2G.

Then M⁰core_G(ker(l))ker (x)ˆ1, so that M is abelian. Hence F is abelian by Step 2.

STEP 4. Let N/G and let u2Irr…N† be under x. Let g2G. Then u^gˆu^sfor somes2Gal…Q…u†=Q†. Alsouis faithful.

LetTˆI_G(u) be the stabilizer ofuinG, and writeTfor the semi-inertia subgroup ofu, this isTˆ fg2Gju^gˆu^s for some s2Gal…Q…u†=Q†g. By part (b) of Lemma 2:2 of [3], ifc2Irr…Tju†is the Clifford correspondent

260 Carolina Vallejo RodrõÂguez

for x, then hˆc^T2Irr…T† inducesx and Q(h)ˆQ(x). By Step 1, we have that TˆG. So that every G-conjugate of u is actually a Galois conjugate. Thus keru^gˆkerufor everyg2G. It follows that ker(u)/G and ker(u) is contained in ker(x) by Clifford's theorem. So u is faithful by Step 1.

STEP5. Ifl2Irr…F†is underx, thenl^Gˆx.

Let l2Irr…F†be underx. Ify2Gis such thatl^yˆl, then we have [x;y]2ker(l) for everyx2F. By step 4, l is faithful, so the element y centralizes F. Since F is self-centralizing, necessarily y2F. We have proved I_G(l)ˆF. This implies l^G is irreducible and thus l^Gˆx. This finishes the proof thatxis monomial.

Now, we work by induction on jGjto show that if U andV are sub- groups of G and l2Irr…U† and m2Irr…V† are linear such that l^Gˆxˆm^G, then the pairs (U;l) and (V;m) are G-conjugate. Since Kˆker(x)core_G(ker(l))\core_G(ker(m)) we may assume that x is faithful, for if K>1 then we can work in G=K. If p is a prime not dividingx(1), thenO_p(G) is contained in bothU andV, becausejG:Uj ˆ x(1)ˆ jG:Vj. By Step 2 (which only required thatxis faithful), we have that

F…G† ˆ Y

pj^ÿx(1)

O_p(G)U\V:

NowlF and m_F are both under x. So that m_Fˆ(lF)^g for someg2Gby Clifford's theorem. We may assume that m_FˆlF ˆn, by replacing the pair (U;l) by some G-conjugate. Thus U and V are contained in TˆI_G(n) and also inT, the semi-interia subgroup ofn. Sincel^Gandm^G are irreducible, also l^T and m^T are irreducible. By uniqueness of the Clifford correspondent, we deduce that l^Tˆm^T. In particular l^T ˆ m^Tˆc2Irr…Tjn†. We know thatQ(c)ˆQ(x), again using Lemma 2:2 of [3]. If T<G, then the result follows by induction. Hence, we may assume TˆG. In particular, arguing as in the first part of the proof, we conclude that n^Gˆx. This implies thatUˆFˆV and the theorem is proven.

Recently we have given a related criterium for monomiality in which the odd degree hypothesis is replaced by certain oddness related to Sylow normalizers (see [4]). This result and our Theorem A seem to be in- dependent. Under the hypothesis of Theorem A, it can also be proved that in fact x issupermonomial, that is, that every character inducing x is monomial.

A criterium for monomiality 261

REFERENCES

[1] R. GOW,Real valued Characters of Solvable Groups, Bull. London Math. Soc., 7 (1975), p. 132.

[2] I. M. ISAACS,Character Theory of Finite Groups, AMS Chelsea Publishing, 2006.

[3] G. NAVARRO- J. TENT,Rationality and Sylow2-subgroups, Proc. Edinburgh Math. Society. 53 (2010), pp. 787-798.

[4] G. NAVARRO- C. VALLEJO,Certain monomial characters, Archiv der Mathe- matik. 99 (2012), pp. 407-411.

Manoscritto pervenuto in redazione il 27 Marzo 2013.

262 Carolina Vallejo RodrõÂguez