A characterization of GVZ groups in terms of fully ramified characters
aa r X i v : . [ m a t h . G R ] J a n A CHARACTERIZATION OF GVZ GROUPS IN TERMS OFFULLY RAMIFIED CHARACTERS
SHAWN T. BURKETT AND MARK L. LEWIS
Abstract.
In this paper, we obtain a characterization of GVZ-groups in termsof commutators and monolithic quotients. This characterization is based oncounting formulas due to Gallagher.
Throughout this paper, all groups are finite. For a group G , we write Irr( G ) forthe set of irreducible characters of G . In this paper, we present a new character-ization of GVZ-groups. A group G is a GVZ-group if every irreducible character χ ∈ Irr( G ) satisfies that χ vanishes on G \ Z ( χ ).The term GVZ-group was introduced by Nenciu in [12]. Nenciu continued thestudy of GVZ-groups in [13] and the second author further continued these studiesin [10]. In our paper [2], we showed that GVZ-groups can be characterized in termsof another class of groups that have appeared in the literature.An element g ∈ G is called flat if the conjugacy class of G is g [ g, G ]. In [14], theydefined a group G to be flat if every element in G is flat. In fact, groups satisfyingthis condition had been studied even earlier. Predating each of these references,Murai [11] referred to such groups as groups of Ono type . In [14], they proved thatif G is nilpotent and flat, then G is a GVZ-group. Improving this result, we provein [2] that a group G is a GVZ-group if and only if it is flat.In this paper, we characterize GVZ-groups using fully ramified characters. Fora normal subgroup N of G , we say that the character χ ∈ Irr( G ) is fully ramified over N if χ N is homogeneous and χ ( g ) = 0 for every element g ∈ G \ N .Following the literature, a group G is called central type if there is an irreduciblecharacter of G that is fully ramified over the center Z ( G ). Results about centraltype groups are in [4], [5], [6], and [7].With this as motivation, we define an irreducible character χ of G to be centraltype if χ , considered as a character of G/ ker( χ ), is fully ramified over Z ( G/ ker( χ )).(I.e., G/ ker χ is a group of central type with faithful character χ .) It is not difficultto see that G is a GVZ-group if and only if every character χ ∈ Irr( G ) is of centraltype.Recall from the literature that a group is called monolithic if it has a uniqueminimal normal subgroup. It is easy to see that if N is a normal subgroup of G and G/N is monolithic, then N appears as the kernel of some irreducible character of G .Also an irreducible character χ is called monolithic if the quotient group G/ ker( χ )is monolithic. Thus, monolithic quotients correspond to monolithic characters.The purpose of this paper is to give a new characterization of central type char-acters based on ideas of Gallagher that are encapsulated in [9, Theorem 1.19 and Date : January 28, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
GVZ groups; p -groups; fully ramified characters. Lemma 1.20], thereby obtaining a new characterizations of GVZ-groups. In partic-ular, we prove the following theorem.
Theorem 1.
Let G be a nonabelian group. Then the following are equivalent: (1) G is a GVZ-group. (2) For every monolithic character χ ∈ Irr( G ) and for every element g ∈ G \ Z ( χ ) , there exists an element x ∈ G so that [ g, x ] ∈ Z ( χ ) \ ker( χ ) . (3) G is nilpotent, and for every normal subgroup N of G for which G/N ismonolithic and for every element g ∈ G satisfying [ g, G ] (cid:2) N , there existsan element x ∈ G such that [ g, x ] / ∈ N and [[ g, x ] , G ] ≤ N . Our proof relies on the following lemma, which we will see is an immediateconsequence of some arguments of Gallagher that can be found in [9, Theorem 1.19and Lemma 1.20]. For an element g ∈ G , we set D G ( g ) = { x ∈ G | [ x, g ] ∈ Z ( G ) } .Observe that D G ( g ) /Z ( G ) = C G/Z ( G ) ( gZ ( G )), so D G ( g ) is always a subgroup of G . Lemma 2.
Let G be a group. If the character ϑ ∈ Irr( Z ( G )) is faithful, then ϑ isfully ramified with respect to G/Z ( G ) if and only if [ g, D G ( g )] = 1 for every element g ∈ G \ Z ( G ) .Proof. By Theorem 1.19 and Lemma 1.20 of [9], the number of irreducible con-stituents of ϑ G equals the number of conjugacy classes of cosets gZ ( G ) ∈ G/Z ( G )that satisfy [ g, D G ( g )] = 1. Observe that if g ∈ Z ( G ), then [ g, D G ( g )] = 1. Hence,the only way that there can be only one conjugacy class of elements of in G/Z ( G )satisfying this condition is if [ g, D G ( g )] = 1 for all elements g ∈ G \ Z ( G ). Since ϑ is fully ramified with respect to G/Z ( G ) if and only if ϑ G has a unique irreducibleconstituent, it follows that ϑ is fully ramified with respect to G/Z ( G ) if and onlyif there is only one conjugacy class satisfying the condition. This gives the desiredresult. (cid:3) We get a slightly stronger statement without much difficulty.
Lemma 3.
Let G be a group. If λ ∈ Irr( Z ( G )) is a character, then λ is fullyramified with respect to G/Z ( G ) if and only if [ g, D G ( g )] (cid:2) ker( λ ) for every element g ∈ G \ Z ( G ) .Proof. Let Z = Z ( G ) and let K = ker( λ ). Suppose first that λ is fully ramifiedwith respect to G/Z . Since λ is fully ramified with respect to G/Z , it followsthat
Z/K = Z ( G/K ). Applying Lemma 2, we have that [ gK, D
G/K ( gK )] = 1for all cosets gK ∈ G/K \ Z/K . It is not difficult to see that this implies that[ g, D G ( g )] (cid:2) K for all elements g ∈ G \ Z . Conversely, suppose that [ g, D G ( g )] (cid:2) K for all g ∈ G \ Z . Hence, we have [ gK, D G/K ( gK )] = 1 for all gK ∈ G/K \ Z/K . Thisimplies that [ gK, G/K ] = 1 for all cosets gK ∈ G/K \ Z/K , and so Z ( G/K ) ≤ Z/K .Since
Z/K ≤ Z ( G/K ) obviously holds, we have Z ( G/K ) =
Z/K . Notice that λ is a faithful character of Z/K , so we may apply Lemma 2 to see that λ is fullyramified with respect to G/KZ/K ∼ = GZ . (cid:3) Let G be a group, fix a character χ ∈ Irr( G ), and write χ Z ( G ) = χ (1) λ forsome character λ ∈ Irr( Z ( G )). Note that ker( λ ) = ker( χ ) ∩ Z ( G ). Consider anelement g ∈ G . Since [ g, D G ( g )] ≤ Z ( G ), we have [ g, D G ( g )] (cid:2) ker( λ ) if and only if[ g, D G ( g )] (cid:2) ker( χ ). Furthermore, [ g, D G ( g )] (cid:2) ker( χ ) if and only if there exists an VZ-GROUPS 3 element x ∈ G so that [ g, x ] ∈ Z ( G ) \ ker( χ ). Hence, Lemma 3 can be equivalentlystated as follows. Lemma 4.
Let G be a group. A character χ ∈ Irr( G ) is fully ramified over Z ( G ) if and only if for every element g ∈ G \ Z ( G ) , there exists an element x ∈ G forwhich [ g, x ] ∈ Z ( G ) \ ker( χ ) . This yields the desired characterization of central type characters.
Theorem 5.
The character χ ∈ Irr( G ) has central type if and only if for everyelement g ∈ G \ Z ( χ ) , there exists an element x ∈ G for which [ g, x ] ∈ Z ( χ ) \ ker( χ ) .Proof. Note that χ is a faithful irreducible character of G/ ker( χ ) and Z ( G/ ker( χ )) = Z ( χ ) / ker( χ ). Thus we see from Lemma 4 that χ , regarded as a characer of G/ ker( χ ), has central type if and only if for every element g ∈ G \ Z ( χ ), thereexists an element x ∈ G for which 1 = [ g, x ] ker( χ ) ∈ Z ( G/ ker( χ )). It is easy tosee that this is equivalent to the statement that was to be proved. (cid:3) Remark . Observe that Theorem 5 implies the well-known result that χ has centraltype if G/Z ( χ ) is abelian (see [8, Theorem 2.31], for example).Before proceeding, we discuss monolithic groups and characters. We need onemore result to prove Theorem 1. This result is proved in our paper [1]. Theorem 7.
The group G is nilpotent if and only if Z ( χ ) > ker( χ ) for eachnonprincipal, monolithic character χ ∈ Irr( G ) . We now prove Theorem 1.
Proof of Theorem 1.
First note the the statement (1) implies (2) follows immedi-ately from Theorem 5.Next we show that (2) implies (3). Let χ ∈ Irr( G ) be monolithic. By Theorem 5, χ has central type. In particular χ (1) = | G : Z ( χ ) | , from which we deduce that Z ( χ ) > ker( χ ) if χ is nonprincipal. Thus G is nilpotent by Theorem 7. Now, let N be a normal subgroup of G for which G/N is monolithic. Then
G/N has a faithfulirreducible character, and thus N = ker( χ ) for some character χ ∈ Irr( G ). Let g ∈ G such that [ g, G ] (cid:2) N . Then gN / ∈ Z ( G/N ) = Z ( χ ) /N and so g / ∈ Z ( χ ). By(1), there exists x ∈ G such that [ g, x ] ∈ Z ( χ ) \ N . Since Z ( χ ) /N = Z ( G/N ), wesee that [[ g, x ] , G ] ≤ N .To complete the proof, we show that (3) implies (1). Fix a prime p that divides | G | , a Sylow subgroup P ∈ Syl p ( G ), and a character ψ ∈ Irr( P ). Consider thecharacter ξ = ψ × H ∈ Irr( G ), where H is a normal p -complement of G . Then G/ ker( ξ ) ∼ = P/ ker( ψ ) is monolithic, by [8, Theorem 2.32]. So ξ is fully ramifiedover Z ( ξ ) = Z ( ψ ) × H by Theorem 5, and this implies that ψ is fully ramified over Z ( ψ ). Now, consider a character χ ∈ Irr( G ). To show that G is a GVZ-group, itsuffices to show that χ is fully ramified over Z ( χ ). Suppose that G = P × · · · × P r is a factorization of G into a direct product of its Sylow subgroups. Then thereexist characters ν i ∈ Irr( P i ) so that χ = ν × · · · × ν r . Observe that Z ( χ ) = Z ( ν ) × · · · × Z ( ν r ). We have already shown that each ν i is fully ramified over Z ( ν i ) and so it follows that χ is fully ramified over Z ( χ ), as desired. This proves(1). (cid:3) SHAWN T. BURKETT AND MARK L. LEWIS
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