aa r X i v : . [ m a t h . G R ] J a n PARTIAL GVZ-GROUPS
SHAWN T. BURKETT AND MARK L. LEWIS
Abstract.
Following the literature, a group G is called a group of centraltype if G has an irreducible character that vanishes on G \ Z ( G ). Motivatedby this definition, we say that a character χ ∈ Irr( G ) has central type if χ vanishes on G \ Z ( χ ), where Z ( χ ) is the center of χ . Groups where everyirreducible character has central type have been studied previously under thename GVZ-groups (and several other names) in the literature. In this paper,we study the groups G that possess a nontrivial, normal subgroup N suchthat every character of G either contains N in its kernel or has central type.The structure of these groups is surprisingly limited and has many aspects incommon with both central type groups and GVZ-groups. Introduction
Throughout this paper, all groups are finite. For a group G , we write Irr( G ) forthe set of irreducible characters of G . Let N be a normal subgroup of G . Take χ to be a character in Irr( G ) and assume that χ N is homogeneous; i.e. that χ N is a multiple of an irreducible character of N . Following the literature, when χ satisfies the condition that χ ( g ) = 0 for every element g ∈ G \ N , we say χ is fullyramified over N . It is well-known that χ being fully ramified over N is equivalentto χ N having a unique irreducible constituent θ and θ G having a unique irreducibleconstituent. This is also equivalent to χ N having a unique irreducible constituent θ with the property that | G : N | = ( χ (1) /θ (1)) . (See Lemma 2.29 and Problem6.3 of [11].)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 ). Central type groupshave been studied extensively in the literature. We mention a few of the importantpapers: [6], [7], [8], and [10]. In particular, it is known that if G is of central type,then G is solvable (see [10]), and G is central type if and only if all of the Sylowsubgroups of G are central type (see [6]).With this as motivation, we say that an irreducible character χ of G has 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 .) Groups whereevery member of Irr( G ) has central type have been called GVZ-groups in the liter-ature (see [14, 17, 18]), and we use this terminology in this paper. Such groups arenecessarily nilpotent (see [17]); thus, it is not difficult to see that G is a GVZ-groupif and only if G is the direct product of GVZ groups of prime power orders fordifferent primes. Moreover, we show in [2] that G is a GVZ-group if and only ifcl G ( g ) = g [ g, G ] for each element g ∈ G . Date : January 28, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
GVZ groups; p -groups; fully ramified characters; flat groups. We have seen that having a central type character whose center equals the centerof the group implies that the group is solvable and that having all irreduciblecharacters be central type implies that the group is nilpotent. It makes sense toask what can be said if some but not necessarily all of the irreducible charactersare known to have central type.When N is a normal subgroup of a group G , we write Irr( G | N ) for the setof irreducible characters of G whose kernels do not contain N . This set was firststudied explicitly by Isaacs and Knutson in [12] where they consider the impact ofa number of conditions on Irr( G | N ) on the structure of N . We now consider theassumption that all the characters in Irr( G | N ) have central type. In particular,we obtain results regarding the structure of G as opposed to just the structure of N . Theorem A.
Let N be a nontrivial, normal subgroup of a group G . Assumethat every character in Irr( G | N ) has central type. Then G is nilpotent, and cl G ( g ) = g [ g, G ] for every element g ∈ N . When we make the additional assumption that N does not have prime powerorder, we obtain a much stronger conclusion. Theorem B.
Let N be a nontrivial, normal subgroup of a group G . If N does nothave prime power order and every character in Irr( G | N ) has central type, then G is a GVZ-group. When N does have prime power order, we also obtain strong information on thestructure of G . Note that since G is nilpotent by Theorem A Q will be normal inthis theorem. Theorem C.
Let p be a prime, and suppose N is a nontrivial, normal subgroupof a group G such that every character in Irr( G | N ) has central type. If N is a p -group and Q is the (normal) Hall p -complement of G , then Q is a GVZ-group. In Theorem B of [2], we show when G is a GVZ-group that the nilpotence classof G is bounded by | cd( G ) | , where cd( G ) is the set of degrees of the irreduciblecharacters of G . We now obtain a similar result regarding the nilpotence class of N when every character in Irr( G | N ) has central type. Theorem D.
Let N be a nontrivial, normal subgroup of a group G . If everycharacter in Irr( G | N ) has central type, then the nilpotence class of N is at most |{ χ (1) | χ ∈ Irr( G | N ) }| . As suggested above, a conjugacy class cl G ( g ) satisfying cl G ( g ) = g [ g, G ] can beconsidered a conjugacy class analog of a central type character. We call such aconjugacy class a flat class of G , or we simply say that the element g is flat in G .We have seen that a group G is nilpotent when each of its elements is flat. Wemention that flat elements and groups where all the elements are flat were firststudied in [19]. We next present a generalization of this fact for normal subgroupssuggested by Theorem A. Theorem E.
Suppose every element in N is flat in G . Then N is contained in Z ∞ ( G ) , the hypercenter of G . In particular, N is nilpotent. Finally, we consider the pairs (
G, N ) where every member of Irr( G | N ) isfully ramified over N . We will see that these groups have very special structure. ARTIAL GVZ-GROUPS 3
Moreover, they provide examples of groups satisfying the premise of Theorem Athat are not GVZ-groups.
Theorem F.
Let N be a normal subgroup of G . If each member of Irr( G | N ) isfully ramified over N , then G is a p -group and N = Z ( G ) . Partial GVZ groups
Suppose that G has a nontrivial, normal subgroup N so that each member ofIrr( G | N ) has central type. In this situation, we say that G is a partial GVZ-groupwith respect to N . In this section, we prove that partial GVZ-groups are nilpotentgroups that are GVZ-groups up to perhaps a single Sylow direct factor. We beginwith a lemma about the intersection of the kernels of the characters in Irr( G | N ). Lemma 2.1.
Let N be a nontrivial, normal subgroup of G . Then the intersectionof the kernels of the characters in Irr( G | N ) is trivial.Proof. Write K = T χ ∈ Irr( G | N ) ker( χ ). Let ψ ∈ Irr( N ) be nonprincipal. Notice that K ≤ ker( χ ) if N (cid:2) ker( χ ). In particular, K ≤ ker( χ ) for each χ ∈ Irr( G | ψ ). Itfollows that K ≤ ker( ψ G ), which is contained in N . But K ∩ N is trivial, as thisis the intersection of the kernels of all of the irreducible characters of G . It followsthat K = 1, as desired. (cid:3) This next lemma appeared as Lemma 3.8 in [1].
Lemma 2.2.
Let N and M be normal subgroups of G . Write Z ( G/N ) = Z N /N and Z ( G/M ) = Z M /M . Then Z ( G/ ( N ∩ M )) = ( Z N ∩ Z M ) / ( N ∩ M ) . We now come to the first main theorem of this section. We consider the Sylowsubgroups of partial GVZ groups.
Theorem 2.3.
Let N be a nontrivial, normal subgroup of G and suppose that everycharacter χ ∈ Irr( G | N ) is a central type character. Let S be a Sylow subgroup of G . Then S ∩ Z ( G ) = Z ( S ) . If ϑ ∈ Irr( S ) and N (cid:2) ker( ϑ ) , then ϑ is a central typecharacter. Furthermore, if S intersects N trivially, then S is a GVZ-group.Proof. To prove the first statement, we essentially use the argument given in theproof of [6, Theorem 2], taking a bit of extra care where necessary. Let S be aSylow subgroup of G , say for the prime p . Let ϑ ∈ Irr( S ) satisfy N (cid:2) ker( ϑ ). Since N (cid:2) ker( ϑ G ), there exists a character χ ∈ Irr( G | N ) lying over ϑ . In particular, χ is a central type character of G . Write χ Z ( χ ) = χ (1) λ , where λ ∈ Irr( Z ( χ )). Let R = Z ( χ ) S . Let γ ∈ Irr( R | ϑ ) such that h χ R , γ i >
0. Note that S ∩ ker( χ ) ≤ ker( ϑ ),so ϑ can be considered a character of S = S ker( χ ) / ker( χ ). Similarly γ can beconsidered a character of R/ ker( χ ). Let X denote the image of X ⊆ G under theprojection G → G/ ker( χ ). Then R = Z ( G ) S = O p ′ ( Z ( G )) × S and so γ S = ϑ .Since γ lies over λ and λ G is homogeneous, γ G = eχ for some integer e . Then e = | G : R | γ (1) /χ (1) = | G : R | ϑ (1) /χ (1). Since | G : R | is not divisible by p , itfollows that χ (1) p = | G : Z ( χ ) | p = | S : S ∩ Z ( χ ) | = | S : S ∩ Z ( G ) | divides ϑ (1) .Since G is a group of central type, S ∩ Z ( G ) = Z ( S ) [6, Theorem 2], from which itfollows that ϑ is a central type character of S .Now we show that S ∩ Z ( G ) = Z ( S ). Let χ ∈ Irr( G | N ). As before, let X denote the image of X ⊆ G under the projection G → G/ ker( χ ). Since G is agroup of central type, we conclude from [6, Theorem 2] that Z ( S ) = S ∩ Z ( G ). SHAWN T. BURKETT AND MARK L. LEWIS
By the modular law, we see that Z ( S ) = S ∩ Z ( G ) = S ∩ Z ( χ ). The isomorphism S → S/ ( S ∩ ker( χ )) therefore implies that Z ( S/ ( S ∩ ker( χ ))) = ( S ∩ Z ( χ )) / ( S ∩ ker( χ )). Writing K = T χ ∈ Irr( G | N ) ker( χ ) and Z = T χ ∈ Irr( G | N ) Z ( χ ), it followsfrom Lemma 2.2 that Z ( S/ ( S ∩ K )) = ( S ∩ Z ) / ( S ∩ K ), as χ ∈ Irr( G | N ) waschosen arbitrarily. Since K = 1 by Lemma 2.1, we conclude from Lemma 2.2 that Z = Z ( G ). Thus S ∩ Z ( G ) = Z ( S ), as required.Finally assume that S ∩ N = 1. Let ψ ∈ Irr( N ) be nonprincipal and let χ ∈ Irr( G | ψ ). Then χ ∈ Irr( G | N ), so χ is a central type character. Then G/ ker( χ )is a group of central type and S ∩ ker( χ ) ≤ S ∩ N = 1. Thus S ker( χ ) / ker( χ ) ∼ = S is a group of central type [6, Theorem 2]. (cid:3) With Theorem 2.3, we will show G must actually be nilpotent, which is the firststatement in Theorem A. Before doing this however, we prove the second statementin Theorem A. In fact, we prove something slightly stronger. Theorem 2.4.
Let N be a normal subgroup of G . If every member of Irr( G | [ N, G ]) has central type, then every element of N is flat in G .Proof. Let g ∈ N . By column orthogonality | C G ( g ) | = X χ ∈ Irr( G | [ g,G ]) | χ ( g ) | + X χ ∈ Irr( G/ [ g,G ]) | χ ( g ) | . Since Irr( G/ [ g, G ]) is exactly the set of irreducible characters χ for which g ∈ Z ( χ ),we see that the second sum is | G/ [ g, G ] | . Note that [ g, G ] ≤ [ N, G ] and so Irr( G | [ g, G ]) ⊆ Irr( G | [ N, G ]). Thus each member of Irr( G | [ g, G ]) is a central typecharacter. Also g / ∈ Z ( χ ) for any member of Irr( G | [ g, G ]). This means the firstsum is zero, which implies | C G ( g ) | = | G/ [ g, G ] | . We deduce that | cl G ( g ) | = | [ g, G ] | ,and the result follows. (cid:3) Next, we provide an example that shows the converse of Theorem 2.4 is false.
Example 2.5.
The converse to Lemma 2.4 is false. Let G be nonnilpotent with N = Z ( G ) > Z ( G ). Then g [ g, G ] = cl G ( g ) for every g ∈ N , but by Theorem Athere must exist a character in Irr( G | [ N, G ]) that is not central type since G isnot nilpotent.Although the converse of Lemma 2.4 is false, we can still determine that N isnilpotent under the assumption that cl G ( g ) = g [ g, G ] for all elements g ∈ N . Infact, N is hypercentral in this case. We will prove this in Theorem E. Proof of Theorem E.
We work by induction on | G | . Let M ≤ N be minimal normal,and let 1 = g ∈ M . We know 1 / ∈ cl G ( g ). Since g [ g, G ] = cl G ( g ), we see that g − / ∈ [ g, G ], and thus, g / ∈ [ g, G ]. Hence, [ g, G ] < h g i G which must be all of M by minimality. Thus, [ g, G ] = 1 and g ∈ Z ( G ). Since M is minimal normal,we conclude that M ≤ Z ( G ). Observe that G/M satisfies the hypotheses of thetheorem, so the inductive hypothesis implies that
N/M ≤ Z ∞ ( N/M ). Since M ≤ Z ( G ), this implies N ≤ Z ∞ ( G ), as desired. (cid:3) Remark . Observe that if N satisfies the premise of Theorem E, then for each g ∈ N and χ ∈ Irr( G | N ), either g ∈ Z ( χ N ) or χ ( g ) = 0. We deduce from Clifford’sTheorem that either g ∈ Z ( ψ ) or P x ∈ G ψ x ( g ) = 0 for each g ∈ N and ψ ∈ Irr( N ).In particular, every G -invariant irreducible character of N has central type. ARTIAL GVZ-GROUPS 5
Since flat elements stay flat after descending to a quotient, we can still guaranteethe nilpotence of N under a slightly weaker hypothesis. Corollary 2.7.
Suppose that every element in N \ Z ( N ) is flat in G . Then N isnilpotent.Proof. Write Z = Z ( N ), and let X denote the image of X ⊆ G under the projection G → G/Z . Since cl G ( g ) = g [ g, G ] for every g ∈ N \ Z , it follows that cl G (¯ g ) = ¯ g [¯ g, G ]for every element ¯ g ∈ N . Therefore, N is nilpotent by Theorem E, which clearlyimplies that N is nilpotent. (cid:3) We now present the proof of Theorem A.
Proof of Theorem A.
The second statement follows immediately from Theorem 2.4.To prove that G is nilpotent, we work by induction on | G | . By Theorem 2.3, Z ( G )is the product the subgroups Z ( S ) as S ranges over all Sylow subgroups of G . Inparticular Z ( G ) >
1. If G is a p -group, we are done. So assume that G is divisibleby at least two distinct primes p and q . Let P be a Sylow p -subgroup of G , and let Q be a Sylow q -subgroup of G . By Theorem 2.3, Z ( P ) and Z ( Q ) are central andtherefore normal in G . Without loss, we may assume that N (cid:2) Z = Z ( P ). ThenIrr( G/Z | N Z/Z ) ⊆ Irr( G | N ) is nonempty. By the inductive hypothesis, G/Z isnilpotent. Since Z ≤ Z ( G ), it follows that G is nilpotent. (cid:3) We have seen that G is nilpotent if all the characters in Irr( G | N ) have centraltype. It is interesting to ask if there are subsets of G that imply G is nilpotent,given each of its elements are flat in G . We have seen that this is not the case foran arbitrary normal subgroup of G , even if it is noncentral. It seems natural to askif having all elements lying outside of N being flat implies that G is nilpotent. Theanswer is also no. As an example, let G be a Frobenius group of order pq , where p and q are primes. In this case, every element lying outside of the Frobenius kernelwill be flat, and G is certainly not nilpotent.This next theorem includes Theorem C from the Introduction. Theorem 2.8.
Let N be a normal subgroup of G . Then all characters in Irr( G | N ) have central type if and only if G = S × Q , where Q is a GVZ-group, S is a Sylowsubgroup of G that intersects N nontrivially, and all characters in Irr( S | S ∩ N ) have central type.Proof. Suppose first that all of the characters in Irr( G | N ) are central type. Let { p , p , . . . , p n } be the set of prime divisors of | G | , and assume that { p , p , . . . , p ℓ } is the set of prime divisors of | N | . Let S i be a Sylow p i -subgroup of G for each i . From Theorem A we see that G is nilpotent, so G is the direct product ofthe S i . Let χ ∈ Irr( G | N ). Then there exist characters ϑ i ∈ Irr( S i ) such that χ = ϑ × ϑ × · · · × ϑ n . Since N (cid:2) ker( χ ), there exists j , 1 ≤ j ≤ ℓ , such that S j ∩ N (cid:2) ker( ϑ j ). Let µ i ∈ Irr( S i ) for each 1 ≤ i ≤ n , i = j . Write µ j = ϑ j and let ξ = µ × µ × · · · × µ n . Then ξ ∈ Irr( G | N ), so ξ is a central type character. Thus Q i µ i (1) = χ (1) = | G : Z ( χ ) | = Q i | S i : Z ( µ i ) | . By comparing p i -parts for each i , we deduce that µ i (1) = | S i : Z ( µ i ) | for each i . Since each µ i , i = j , was chosenarbitrarily, it follows that each S i for i = j is a GVZ-group. Thus Q = Q i = j S i isalso a GVZ-group, and G = S j × Q . Finally, let ϑ ∈ Irr( S j | S j ∩ N ). Then ϑ liftsto a character ˜ ϑ ∈ Irr(
G/Q | N Q/Q ). Since N (cid:2) ker( ˜ ϑ ), ˜ ϑ has central type. Thus SHAWN T. BURKETT AND MARK L. LEWIS ϑ (1) = ˜ ϑ (1) = | G : Z ( ˜ ϑ ) | = | S j : Z ( ϑ ) | , from which it follows that ϑ has centraltype. This completes the proof of the forward direction.Now, suppose that G = S × Q , Q is a GVZ-group, S is a Sylow subgroup forwhich S ∩ N >
1, and all characters in Irr( S | S ∩ N ) have central type. Suppose χ ∈ Irr( G | N ). We can write χ = ϑ × γ where ϑ ∈ Irr( S ) and γ ∈ Irr( Q ).Notice that χ N = γ (1) ϑ N and so ϑ ∈ Irr( S | S ∩ N ). It is not difficult to see that Z ( χ ) = Z ( ϑ ) × Z ( γ ). We know, by hypothesis, that ϑ is fully ramified with respectto Z ( ϑ ) and γ is fully ramified with respect to Z ( γ ). It is not difficult to see that χ is fully ramified with respect to Z ( χ ). (cid:3) As a corollary, we obtain Theorem B from the Introduction.
Corollary 2.9.
Let N be a normal subgroup of G . If all characters in Irr( G | N ) have central type and N is not a p -group for any prime p , then G is a GVZ-group.Proof. We first appeal to Theorem A to see that G is nilpotent. Since N is nota p -group for any prime p , we can find distinct primes p and p that divide | N | .For i = 1 ,
2, take N i to be the Sylow p i -subgroup of N . Since G is nilpotent, N isnilpotent; so N i is normal and thus characteristic in N and hence, they are normalin G . Using Theorem 2.8 with N i as the normal subgroup, we see that all the Sylowsubgroups of G are GVZ-groups. (We use N to see that Sylow p -subgroup is aGVZ-group and N for the Sylow p -subgroup.) Now, we know that G is nilpotentand that all the Sylow subgroups are GVZ-groups, and so, we can conclude that G is a GVZ-group. (cid:3) We now present the proof of Theorem D, which is an adaptation of the usualTaketa argument. Our proof is nearly identical to our proof of Theorem B of [2]with just a few subtle differences. When G is a nilpotent group, we write c ( G ) forthe nilpotence class of G , and we set G = G and G i +1 = [ G i , G ] for i ≥ G . Following [12], we set cd( G | N ) = { χ (1) | χ ∈ Irr( G | N ) } . Using this notation, we restate Theorem D in a slightly differentform. Theorem 2.10.
Let N be a nontrivial, normal subgroup of G . If all of the char-acters in Irr( G | N ) have central type, then c ( N ) ≤ | cd( G | N ) | .Proof. We work by induction on | G | . If N is abelian, then the result is trivial. Thus,we may assume that N is not abelian, which implies that G is not abelian. Let d < · · · < d n be the distinct degrees in cd( G | N ). Consider a character χ ∈ Irr( G | N ). If ker( χ ) >
1, then | G/ ker( χ ) | < | G | , and cd( G/ ker( χ ) | N ker( χ ) / ker( χ )) ⊆ cd( G | N ). By induction, we have N n ≤ ker( χ ). Thus, if G does not have afaithful character in Irr( G | N ), then N n ≤ ∩ χ ∈ Irr( G | N ) ker( χ ) = 1 by Lemma 2.1.Therefore, we may assume that there exists χ ∈ Irr( G | N ) with ker( χ ) = 1. Thisimplies that Z ( χ ) = Z ( G ). We have d i ≤ | G : Z ( G ) | = χ (1) for every integer i with 1 ≤ i ≤ n . Thus, χ (1) = d n . Notice that if a ∈ cd( G/Z ( G ) | N Z ( G ) /Z ( G )),then a < | G : Z ( G ) | = d n . It follows that | cd( G/Z ( G ) | N Z ( G ) /Z ( G )) | ≤ n − N n − ≤ Z ( G ). It follows that N n − ≤ Z ( G ) ∩ N ≤ Z ( N ). This implies that c ( N ) ≤ n as desired. (cid:3) In Theorem 6.3 of [9], Gagola has proven that if Q is any p -group, then thereexists a p -group P so that Q is isomorphic to a subgroup of P/Z ( P ) and everycharacter in Irr( P | Z ( P )) is fully ramified with respect to Z ( P ). Notice that this ARTIAL GVZ-GROUPS 7 will imply that every irreducible character in Irr( P | Z ( P )) will have central typeand cd( P | Z ( P )) = {| P : Z ( P ) | / } , so | cd( P | Z ( P )) | = 1. Since Q is abitrary,this implies that we have no hope of bounding c ( G ) in terms of | cd( G | N ) | whenevery character in Irr( G | N ) has central type.Even if we assume that G is a GVZ-group, we cannot hope to bound the nilpo-tence class of G in terms of | cd( G | N ) | for a subgroup N . To see this, considerthe groups found in Example 3 in [14]. The groups in that example are all GVZ-groups and all satisfy that | cd( G | Z ( G )) | = 1, but can be chosen to have whatevernilpotence class desired. 3. The subgroup R ( G )Suppose that G is a partial GVZ-group with respect to each of the normalsubgroups N and M . It is not difficult to see that Irr( G | N M ) = Irr( G | N ) ∪ Irr( G | M ), so G is a partial GVZ-group with respect to N M . In particular, there is aunique largest normal subgroup R ( G ) such that every member of Irr( G | R ( G )) hascentral type. In this section, we discuss some properties of this subgroup. Lemma 3.1.
Let G be a group. The following statements are true: (1) R ( G ) = G if and only if G is a GVZ-group; (2) If R ( G ) > , then G = S × Q , where Q is a GVZ-group and S is eithertrivial or a Sylow p -subgroup satisfying < R ( S ) = R ( G ) < S for someprime p ; (3) R ( G/R ( G )) = 1 .Proof. Statement (1) is obvious from the definition, and statement (2) follows im-mediately from Theorem B and Theorem C. So we only show (3). If R ( G ) = 1,then (3) is clear. So assume that R ( G ) >
1, and write
L/R ( G ) = R ( G/R ( G )).Let χ ∈ Irr( G | L ). If R ( G ) ≤ ker( χ ), then χ has central type since L/R ( G ) = R ( G/R ( G )). If R ( G ) (cid:2) ker( χ ), then χ has central type by the definition of R ( G ).Thus L ≤ R ( G ), from which it follows that L = R ( G ). (cid:3) We next look at R ( G ) when G is a direct product. Lemma 3.2.
Let M and N be groups. Assume that M is not a GVZ-group. Thefollowing statements are true: (1) If N is not a GVZ-group, then R ( M × N ) = 1 ; (2) If N is a GVZ-group, then R ( M × N ) = R ( M ) .Proof. We prove the contrapositive of statement (1). Write R = R ( M × N ). Assumethat R >
1. We want to show that that N is a GVZ-group. Since the result isimmediate if N is abelian, we may assume that N is nonabelian. Fix the character ν ∈ Irr( N ), so that ν is nonlinear. We claim that there exists a character µ ∈ Irr( M )so that χ = µ × ν ∈ Irr( M × N | R ). To see this, assume on the contrary that R ≤ ker( µ × ν ) for every character µ ∈ Irr( M ). Consider an element mn ∈ R ,where m ∈ M and n ∈ N . Then mn ∈ ker(1 M × ν ) = M × ker( ν ), so n ∈ ker( ν ).So µ ( m ) ν ( n ) = µ ( m ) ν (1) = µ (1) ν (1) for every character µ ∈ Irr( M ), from whichit follows that m ∈ T µ ∈ Irr( M ) ker( µ ) = 1. Thus we see that R ≤ ker( ν ) ≤ N .Since M is not a GVZ-group, we see that M is not abelian. In particular,Irr( M | M ′ ) is nonempty. Fix a character µ ∈ Irr( M | M ′ ) and consider a character θ ∈ Irr( N | R ). We obtain χ = µ × θ ∈ Irr( M × N | R ). Fix an element SHAWN T. BURKETT AND MARK L. LEWIS m ∈ M \ Z ( µ ). This implies m / ∈ Z ( χ ), so 0 = χ ( m ) = µ ( m ) θ (1) and it followsthat µ ( m ) = 0. We deduce that µ has central type, and since µ was arbitrary, weconclude that M is a GVZ-group, which is a contradiction. Thus, we may find acharacter µ ∈ Irr( M ) so that χ = µ × ν ∈ Irr( M × N | R ), as claimed.Consider elements m ∈ Z ( µ ) and n ∈ N \ Z ( ν ). Since Z ( χ ) = Z ( µ ) × Z ( ν ), wedetermine that mn / ∈ Z ( χ ). Since χ has central type, we calculate 0 = χ ( mn ) = µ ( m ) ν ( n ). Because µ ( m ) = 0, we must have ν ( n ) = 0. Thus, ν has central typeand since ν ∈ Irr( N | N ′ ) was chosen arbitrarily, N is a GVZ-group.Now we show conclusion (2). Assume that N is a GVZ-group. Let R = R ( M × N ). We prove that R ∩ N must be trivial. Suppose R ∩ N >
1. We claim thatthis will imply that M must be a GVZ-group which is contradiction. As we haveseen, if M is abelian, then M will be a GVZ-group, so we assume that M is notabelian. Since R ∩ N >
1, we can find a character 1 = γ ∈ Irr( R | N ). Considercharacters ν ∈ Irr( N | γ ) and µ ∈ Irr( M | M ′ ). Since γ = 1, we see that χ = ν × µ ∈ Irr( N × M | R ( G )). for if not the above argument shows that We claim that R ∩ M isnontrivial. Assume, on the contrary, that R ∩ M is trivial. Let m ∈ M , n ∈ N suchthat mn ∈ R . Since R ⊳ G , it follows that [ m, M ][ n, N ] ⊆ ( M ∩ R )( N ∩ R ) = 1.So m ∈ Z ( M ) and n ∈ Z ( N ). In particular mn ∈ Z ( G ) and h mn i is a normalsubgroup of G . Since mn ∈ R , χ ∈ Irr( M × N | h mn i ) has central type. Let µ ∈ Irr( M | h m i ). Then mn / ∈ ker( µ × N ), so µ × N has central type. Butthis implies that µ has central type. So m ∈ R ( M ). Since m ∈ Z ( M ) ≤ Z ( G ), h m i is also a normal subgroup of G . Since N is a GVZ-group and h m i ≤ R ( M ),every member of Irr( M × N | h m i ) has central type, which contradicts the factthat R ∩ M = 1. So R ∩ M >
1, as claimed. Since R ∩ N = 1, this implies that R ≤ M . So µ × ν ∈ Irr( M × N | R ) if and only if µ ∈ Irr( M | R ). Since N is aGVZ-group, µ × ν has central type if and only if µ does. From this it follows that R ( M ) ≤ R , since every member of Irr( M × N | R ( M )) must have central type, andthat R ≤ R ( M ), since every member of Irr( M | R ) must have central type. Thus R = R ( M ), as desired. (cid:3) Recall that the vanishing-off subgroup V ( χ ) of a character χ is the smallestnormal subgroup V such that χ vanishes on G \ V (e.g. see [11, Chapter 12]). It isnot difficult to see that V ( χ ) = h g ∈ G | χ ( g ) = 0 i and that Z ( χ ) ≤ V ( χ ). Sincecentral type characters vanish off their centers, these are exactly the characterswhose vanishing-off subgroup coincide with their center. Lemma 3.3.
Let V = { χ ∈ Irr( G ) | V ( χ ) > Z ( χ ) } . Then R ( G ) = T χ ∈V ker( χ ) .Proof. Let W = T χ ∈V ker( χ ). If χ ∈ Irr( G | W ), then V ( χ ) = Z ( χ ) and thereforehas central type. So W ≤ R ( G ).Since every member of Irr( G | R ( G )) satisfies V ( χ ) = Z ( χ ), V ⊆
Irr(
G/R ( G )).So R ( G ) ≤ W as well, which completes the proof. (cid:3) As a corollary, we see that every member of Irr( G | R ( G )) is nonlinear, unless G is a GVZ-group. Lemma 3.4. If G is not a GVZ-group, then R ( G ) < G ′ .Proof. Let V = { χ ∈ Irr( G ) | V ( χ ) > Z ( χ ) } . Since G is not a GVZ-group, V is not empty. Let χ ∈ V . For each λ ∈ Irr( G | G ′ ), we have χλ ∈ V . Since R ( G ) = T ξ ∈V ker( ξ ) by Lemma 3.3, we see that R ( G ) ≤ ker( χλ ) ∩ ker( χ ) ≤ ker( λ ) ARTIAL GVZ-GROUPS 9 for each λ ∈ Irr( G | G ′ ). Thus R ( G ) ≤ G ′ . Notice that if R ( G ) = G ′ , then everynonlinear irreducible character of G would be central type and that would implythat G is a GVZ-group, a contradiction. Thus, we must have R ( G ) < G ′ . (cid:3) We next present a result that allows us to find examples of partial GVZ-groupsthat are not GVZ-groups. To do this, we consider a characteristic subgroup that isdefined in Section 6 of [1]. To define it, we begin by defining U ( G ) to be the largestnormal subgroup U of G such that every member of Irr( G | U ) is fully ramified over Z ( G ). By defining U ( G ) /U ( G ) = U ( G/U ( G )), etc, we define an ascending seriesof subgroups U i ( G ) and let U ∞ ( G ) denote its terminal member. Lemma 3.5.
Let N = U ∞ ( G ) . Then every character in Irr( G | N ) has centraltype. In particular, N ≤ R ( G ) .Proof. We work by induction on | G | . Let χ ∈ Irr( G | N ). Suppose that N > U ( G ).By the inductive hypothesis, every character χ ∈ Irr(
G/U | N/U ) has central type.Thus we may assume that χ ∈ Irr( G | U ). By definition, V ( χ ) = Z ( G ) and so χ vanishes on G \ Z ( G ). In particular, χ has central type. Thus every character inIrr( G | N ) has central type, as required. (cid:3) As a consequence of Lemma 3.5, a non GVZ-group G with U ∞ ( G ) > G is not a GVZ-group and U ( G ) >
1, then G is a partial GVZ-group with respect to U ( G ). Thesmallest 2-group with this property has order 128. One may readily check that if G is the group SmallGroup (128, 71) from MAGMA’s Small Groups Library, then1 < U ( G ) = R ( G ). Moreover, one may even find groups G where 1 < U ( G )
We conclude with a special example of a partial GVZ-group. Recall that thesubgroup U ( G ) is defined in [1] to be the largest normal subgroup U of G suchthat every character χ ∈ Irr( G | U ) is fully ramified over Z ( G ). It is easy to seethat U ( G ) ≤ Z ( G ) and that G is a partial GVZ-group with respect to U ( G ) when U ( G ) >
1. In the case that U ( G ) = Z ( G ), the pair ( G, Z ( G )) is an example of a Camina pair .The study of Camina pairs began in [3] where Camina was looking at conditionsthat generalize both Frobenius groups and extra-special p -groups. If G is a groupand N is a normal subgroup of G , we say ( G, N ) is a
Camina pair if every element g ∈ G \ N is conjugate to the entire coset gN . There are a number of equivalentformulations of this condition. For this paper, the most relevant is that ( G, N ) isa Camina pair if and only if each nonprincipal irreducible character of N induceshomogeneously to G .Suppose that ( G, N ) is a Camina pair, and let χ ∈ Irr( G | N ). Then for anyirreducible constituent ψ of χ N , ψ G = eχ for some integer e called the ramificationindex . The integer e satisfies 1 ≤ e ≤ | G : N | . The condition e = 1 is equivalent to ψ G ∈ Irr( G ), and the condition e = | G : N | is equivalent to χ being fully ramifiedover N . It is well-known that every nonprincipal irreducible character of a subgroup N induces irreducibly to G if and only if G is a Frobenius group and N = F ( G ).Thus a Frobenius group is a Camina pair ( G, N ) where the ramification index ofeach χ ∈ Irr( G | N ) is as small as possible. Observe that Theorem F provides acharacterization of those Camina pairs ( G, N ) where the ramification index of each χ ∈ Irr( G | N ) is as large as possible.Before proving Theorem F, we motivate the conclusion a bit. Let ( G, N ) be aCamina pair. Then it is easy to see that Z ( G ) ≤ N , and we say ( G, N ) is a centralCamina pair in the extreme case that N = Z ( G ). Central Camina pairs are thefocus of [15]. Note that ( G, Z ( G )) is a Camina pair if and only if U ( G ) = Z ( G ),which happens if and only if ( G, U ( G )) is a Camina pair. It is known that if ( G, N )is a central Camina pair, then G is a p -group and every character in Irr( G | N ) isfully ramified over N . Thus Theorem F provides a sort of converse to this statement. Proof of Theorem F.
We prove this by induction on | G | . First, we show that G must be a p -group. Since every character in Irr( G | N ) vanishes on G \ N , wesee that ( G, N ) is a Camina pair, and that G cannot be a Frobenius group withFrobenius kernel N . So either N is a p -group, or G/N is a p -group [3, Theorem 2.2].If N is a p -group, then it follows from [13, Theorem 5.1] that I G ( ψ ) is a p -group foreach nonprincipal ψ ∈ Irr( N ). Since every nonprincipal irreducible character of N is fully ramified with respect to G/N , this would force G to be a p -group as well.So we assume that G/N is a p -group.Since | G : N | divides χ (1) for every character χ ∈ Irr( G | N ), we know that p divides χ (1) for every character χ ∈ Irr( G | N ′ ) if N ′ >
1. It follows from [12,Theorem D] that N (and hence also G ) is solvable. Since every character ψ ∈ Irr( N )is invariant in G , we see that [ N, G ] = N ′ . Suppose that 1 < M ≤ N ′ is a normalsubgroup of G . Then ( G/M, N/M ) is a Camina pair by the inductive hypothesis.Since every nonprincipal character ψ ∈ Irr(
N/M ) is fully ramified with respectto
G/N , it follows from the inductive hypothesis that
N/M = Z ( G/M ). Thus, N ′ = [ N, G ] ≤ M , and we deduce that N ′ is a minimal normal subgroup of G .Therefore, N ′ is an elementary abelian q -group for some prime q . Also, note that N/N ′ is an elementary abelian p -group [16, Theorem 2.2]. If q = p , then we aredone, so assume q = p . Then N ′ is a normal p -complement for G , and P ∼ = G/N ′ .Since Z ( G/N ′ ) = N/N ′ , we have | Z ( P ) | = | N : N ′ | = | N ∩ P | . But Z ( P ) ≤ N by[4, Proposition 3.4], which allows us to conclude Z ( P ) = P ∩ N . By [5, Theorem3], G must be a p -group, a contradiction. Therefore q = p and G is a p -group, asclaimed.Since N ′ is a minimal normal subgroup of G , it follows that N ′ ≤ Z ( G ). Byapplying the Three-subgroups lemma, we deduce that [ G ′ , N ] = 1. Since N ≤ G ′ ,this implies N ≤ Z ( G ′ ), which in turn implies that N is abelian. In particular, ARTIAL GVZ-GROUPS 11 this yields 1 = N ′ = [ N, G ], from which we conclude N ≤ Z ( G ). Since ( G, N ) is aCamina pair, we also have Z ( G ) ≤ N , which completes the proof. (cid:3) Of course, we get the desired characterization of central Camina pairs as a corol-lary. Observe the similarity to the aforementioned characterization of Frobeniusgroups.
Corollary 4.1.
The group G is a central Camina pair if and only if G has a normalsubgroup N such that every character χ ∈ Irr( G | N ) is fully ramified over N . Inthis case, N = Z ( G ) . References [1] S. T. Burkett and M. L. Lewis, Groups where the centers of the irreducible characters forma chain II, Monatsch. Math. 192 (2020), 783-812.[2] S. T. Burkett and M. L. Lewis, GVZ groups, flat groups, and CM groups, preprint.[3] A. R. Camina, Some conditions which almost characterize Frobenius groups, Israel J. Math.31 (1978), 153-160.[4] D. Chillag and I. D. MacDonald, Generalized Frobenius groups, Israel J. Math. 47 (1984),111-122.[5] D. Chillag, A. Mann, and C. M. Scoppola, Generalized Frobenius groups. II, Israel J. Math.62 (1988), 269–282.[6] F. R. DeMeyer and G. J. Janusz, Finite groups with an irreducible character of large degree,Math. Z. 108 (1969), 145-153.[7] A. Espuelas, On certain groups of central type, Proc. Amer. Math. Soc. 97 (1986), 16-18.[8] S. M. Gagola, Jr., Characters fully ramified over a normal subgroup, Pacific J. Math. 55(1974), 107-126.[9] S. M. Gagola, Jr., Characters vanishing on all but two conjugacy classes. Pacific J. Math.109 (1983), 363-385.[10] R. B. Howlett and I. M. Isaacs, On groups of central type, Math. Z. 179 (1982), 552-569.[11] I. M. Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994.[12] I. M. Isaacs and G. Knutson, Irreducible character degrees and normal subgroups, J. Algebra199 (1998), 302–326.[13] E. B. Kuisch and R. W. van der Waall, Homogeneous character induction, J. Algebra 149(1992), 454–471.[14] M. L. Lewis, Groups where the centers of the irreducible characters form a chain, Monatsh.Math. 192 (2020), 371–399.[15] M. L. Lewis, On p -group Camina pairs, J. Group Theory 15 (2012), 469—483.[16] I. D. Macdonald, Some p -groups of Frobenius and extra-special type, Israel J. Math. 40(1981), 350–364.[17] A. Nenciu, Isomorphic character tables of nested GVZ-groups, J. Algebra Appl. 11 (2012),1250033, 12 pp.[18] A. Nenciu, Nested GVZ-groups, J. Group Theory 19 (2016), 693-704.[19] H. Tandra, and W. Moran, Flatness conditions on finite p -groups, Comm. Algebra, 32 (2004),2215–2224. Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242,U.S.A.
Email address : [email protected] Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242,U.S.A.
Email address ::