J. Group Theory 19 (2016), 763–776 DOI 10.1515/jgth-2016-0010 © de Gruyter 2016 Finite groups with coprime character degrees and codegrees Dengfeng Liang and Guohua Qian Communicated by Gunter Malle G ker Abstract. We investigate the structure of finite groups G in which gcd..1/; j .1/W j / 1 for all irreducible characters of G. D 1 Introduction In this paper, G always denotes a finite group, all characters are complex char- acters, and we use Isaacs [4] as a source for standard notation and results from character theory. For a character of G, the second author [8, 11] defined its codegree by G ker c.1/ j W j: D .1/ Clearly, if is irreducible, then c.1/ is an integer divisor of G= ker . Gagola j j and Lewis [1] showed that G is nilpotent if and only if .1/ divides c.1/ for every Irr.G/. The second author [10] obtained a p-local version of the above 2 result for an odd prime p, and proved that G has a normal Sylow p-subgroup if c and only if .1/p < .1/p for every Irr.G/ with p dividing .1/, where mp 2 denotes the p-part of an integer m. Opposite to the case when .1/ divides c.1/, in this paper we will study another extreme case, namely, gcd..1/; c.1// 1 D for all Irr.G/. 2 An integer m is called a Hall number with respect to a finite group G, pro- vided that m divides G and gcd.m; G =m/ 1. If all of the irreducible character j j j j D degrees of G are Hall numbers with respect to G, then G is called an H1-group. A character of G is called an H-character if .1/ is a Hall number with respect to G= ker , i.e., gcd..1/; c.1// 1; and G is called an H-group if all its irre- D ducible characters are H-characters. Clearly all linear characters are H-characters, and all H1-groups are H-groups. Note that H1-groups have been studied in [5]. Project supported by the Nature Science Foundation of China (No. 11471054 and 11501017). The corresponding author is Guohua Qian. 764 D. Liang and G. Qian We say that a group A acts Frobeniusly on a group B if A Ë B is a Frobenius group with kernel B and complement A. A group G is said to be a 2-Frobenius group if there are G-invariant subgroups M and N such that G=N and M are Frobenius groups with M=N and N , respectively, as their Frobenius kernels, and such a 2-Frobenius group is denoted by Frob2.G; M; N /. Let F1.G/ F .G/, D and let Fi .G/ be the ith ascending Fitting subgroup of G. The nilpotent length (or Fitting height) of a solvable group G, denoted by nl.G/, is the smallest number l such that F .G/ G. l D Theorem A. Let G be a solvable group. Then G is an H-group if and only if one of the following holds: (1) G is an abelian group. (2) G M Ë F.G/, where M is cyclic, F.G/ is an abelian Hall subgroup of G, D and every Sylow subgroup P of M acts Frobeniusly on ŒF.G/; P . (3) G L Ë F2.G/ satisfies the following conditions: D (3.1) L is cyclic and has square-free order, and L and F2.G/=F.G/ are coprime. j j j j (3.2)F 2.G/ is of type (2). (3.3) If a prime p divides both L and F.G/ , then j j j j p p O 0 .G/ Frob2.O 0 .G/; M; N /; D where Op0 .G/=M p and, for a positive integer e, N pep and j j D j j D M=N .pep 1/=.pe 1/. j j D By Theorem A, a solvable H-group has nilpotent length at most 3. Note that a solvable H-group does not need to be an H1-group, the symmetric group of degree 4 is an example; however, a solvable H-group with nilpotent length at most 2 must be an H1-group, see Corollary 2.4 below. Let Sol.G/ be the largest solvable normal subgroup of G, and GS the smallest normal subgroup of G such that G=GS is solvable. For a G-invariant subgroup N , we write Irr.G N/ Irr.G/ Irr.G=N /, cd.G N/ .1/ Irr.G N/ . j D j D ¹ W 2 j º Theorem B. Let G be a nonsolvable group. Then G is an H-group if and only if G satisfies the following conditions: (1) G D Ë .L N/, where D; L and N are Hall subgroups of G. D (2) L GS , L=Sol.L/ PSL.2; 2f / for some integer f 2, and we have either D Š Sol.L/ 1 or Sol.L/ 22f and cd.L Sol.L// 22f 1 . D j j D j D ¹ º (3) D is cyclic and D is a square-free divisor of f . j j (4) G=L is a solvable H-group. Finite groups with coprime character degrees and codegrees 765 We construct examples showing the existence of nonsolvable H-groups that are f not H1-groups. Let F be a field with 2 elements, let ˛ be a field automorphism of F such that o.˛/ is square-free and coprime to 2f .22f 1/, and let V be a vector space of dimension 2 over F . Then SL.2; 2f / acts naturally on V and ˛ acts naturally on SL.2; 2f /. Note that ˛ also acts on the elementary abelian 2- group V in the following way, ˛.k1u k2v/ ˛.k1/u ˛.k2/v; C D C where k1; k2 F , and u; v constitute a base of the vector space V . Write 2 L SL.2; 2f / Ë V and G ˛ Ë L: D D h i Then G is a group. It is known that cd.L V/ 22f 1 (see [12]). Now one can j D ¹ º check directly that both L and G are H-groups (see also the proof of Theorem B for details), but neither is an H1-group. Let N E G, Irr.N /, and be a character of G. As usual, we denote by 2 IG./ the inertia group of in G, and we denote by Irr./ the set of irreducible constituents of . 2 Theorem A We begin with a list of some fundamental results about character codegree, which will be often used freely in the sequel. Lemma 2.1. Let G be a finite group and Irr.G/. 2 (1) Let N be a G-invariant subgroup of ker . Then may be viewed as an irreducible character of G=N , and the codegrees of in G and in G=N are the same. c (2) If M is subnormal in G and is an irreducible constituent of M , then .1/ divides c.1/. (3) If is induced by an irreducible character of a subgroup of G, then c.1/ divides c.1/. (4) If p is a prime divisor of G , then p divides c.1/ for some Irr.G/. j j 2 (5) If G is an H-group, then M=N is also an H-group whenever M is subnormal in G and N is normal in M . Proof. Statements (1) and (2) follow from [11, Lemma 2.1]. Statement (3) is [9, Lemma 2.2], and (4) is [11, Theorem A]. (5) Let Irr.M=N / and Irr. G/. Then .1/ .1/ by [4, Lemma 6.8], 2 2 j and c.1/ c.1/ by (2), so M=N is also an H-group. j 766 D. Liang and G. Qian Lemma 2.2 ([5, Theorem 1.1]). Let G be a solvable group. Then G is an H1-group if and only if one of the following holds: (1) G is abelian. (2) G M Ë F , where M is a cyclic Hall subgroup of G, F is an abelian Hall D subgroup of G, and if ŒF; P > 1, then P acts Frobeniusly on ŒF; P for every Sylow subgroup P of M . (3) G L Ë D, where L is a cyclic Hall subgroup of G and of square-free order, D D is of type (2). Lemma 2.3. Let G be a solvable H-group. (1) If G is nilpotent, then G is abelian. (2) If nl.G/ 2, then G A Ë F.G/, where A is cyclic, F.G/ is an abelian Hall D D subgroup of G. Furthermore, every Sylow subgroup P of A acts Frobeniusly on ŒF.G/; P . Proof. (1) Observe that .1/ divides c.1/, that is, .1/2 divides G=ker for j j every irreducible character of a nilpotent group G. We get (1). (2) Write F.G/ F . Note that G=F is nilpotent, by (1) and Lemma 2.1, F and D G=F are abelian. To see the rest of the statement, we may assume by induction and Lemma 2.1 that G PF > F , where P Syl .G/ for some prime p. Now D 2 p we need to show that P F 1, or equivalently F is a p -group, and that P acts \ D 0 Frobeniusly on ŒF; P . Assume P F > 1 and write F .P F/ U , where U Op .F /.