Zeros of primitive characters of finite groups

Sesuai Yash Madanha

doi:10.1515/jgth-2019-2051

Article Publicly Available

Zeros of primitive characters of finite groups

Sesuai Yash Madanha

Published/Copyright: December 4, 2019

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From the journal Journal of Group Theory Volume 23 Issue 2

Abstract

We classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside’s classical theorem on zeros of characters.

1 Introduction

This work is a continuation of what we began in [16], the classification of finite non-solvable groups with a faithful primitive complex irreducible character that vanishes on exactly one conjugacy class. In [16], we completed the work for groups with a non-abelian composition factor isomorphic to a sporadic simple group, an alternating group, An, n≥5, or a special linear group PSL2⁢(q), q≥4. In this article, we finish the classification for all finite non-solvable groups. This is a contribution to the more general problem of the classification of finite groups with an irreducible character that vanishes on exactly one conjugacy class (see [26, 7, 23, 3, 16] for more work on this problem). We begin by looking at the following problem.

Problem 1.

For each quasisimple group M, classify all faithful complex irreducible characters χ such that there exists some prime p such that

χ vanishes on elements of the same p-power order,
the number of conjugacy classes that χ vanishes on is at most the size of the outer automorphism group of the group M/Z⁢(M),
Z⁢(M) is cyclic and of p-power order.

For convenience, we shall define the following property for a finite group M. We shall say (*) holds for M if

(*)a faithful irreducible character⁢χ⁢of⁢M⁢has properties (i)–(iii) of Problem 1.

We completely solve Problem 1 in this article.

Theorem 1.1.

Let M be a quasisimple group. If (*) holds for M, then M is one of the following:

M=PSL2⁢(5), χ⁢(1)=3 or χ⁢(1)=4,
M=SL2⁢(5), χ⁢(1)=2 or χ⁢(1)=4,
M=3⋅A6, χ⁢(1)=9,
M=PSL2⁢(7), χ⁢(1)=3,
M=PSL2⁢(8), χ⁢(1)=7,
M=PSL2⁢(11), χ⁢(1)=5 or χ⁢(1)=10,
M=PSL2⁢(q), χ⁢(1)=q, where q≥5,
M=PSU3⁢(4), χ⁢(1)=13,
M=B22⁢(8), χ⁢(1)=14.

Using Theorem 1.1, we classify finite non-solvable groups with a faithful primitive irreducible character that vanishes on one conjugacy class. We showed in [16] that it is sufficient to only consider automorphism groups of the groups in Theorem 1.1.

Theorem 1.2.

Let G be a finite non-solvable group. Then χ∈Irr⁢(G) is faithful, primitive and vanishes on one conjugacy class if and only if G is one of the following groups:

G=PSL2⁢(5), χ⁢(1)=3 or χ⁢(1)=4,
G=SL2⁢(5), χ⁢(1)=2 or χ⁢(1)=4,
G∈{A6:22,A6:23,3⋅A6:23}, χ⁢(1)=9 for all such G,
G=PSL2⁢(7), χ⁢(1)=3,
G=PSL2⁢(8):3, χ⁢(1)=7,
G=PGL2⁢(q), χ⁢(1)=q, where q≥5,
G=B22⁢(8):3, χ⁢(1)=14.

The result below follows easily.

Corollary 1.3.

Let G be a finite non-abelian simple group, and let χ∈Irr⁢(G). If χ vanishes on exactly one conjugacy class, then one of the following holds:

G=PSL2⁢(5), χ⁢(1)=3,
G=PSL2⁢(7), χ⁢(1)=3,
G=PSL2⁢(2a), χ⁢(1)=2a, where a≥2.

Corollary 1.3 positively answers a question posed by Dixon and Rahnamai Barghi [7, Remark 11].

We now look at what our results imply with regards to the classical theorem of Burnside on zeros of characters. There have been some generalisations of Burnside’s theorem (see [18, 1, 22]).

Burnside’s theorem can be rewritten as follows.

Theorem 1.4 (Burnside’s theorem).

Let G be a finite group, and let χ∈Irr⁢(G). If χ⁢(1) is divisible by a prime, then χ vanishes on at least one conjugacy class.

Using [7, Propositions 1 (i) and 4] and Theorem 1.2, we have the following theorem.

Theorem 1.5.

Let G be a finite group whose non-abelian composition factors are not isomorphic to B22⁢(8). Let χ∈Irr⁢(G) be primitive. If χ⁢(1) is divisible by two distinct prime numbers, then χ vanishes on at least two conjugacy classes.

The non-solvable group B22⁢(8):3 is a real exception by Theorem 1.2. Hence this extends Burnside’s theorem when the character is primitive. It would be interesting to know if the primitivity is necessary for Theorem 1.5 to hold.

2 Preliminaries

In this section, we present some results we will need to use. A lot of work on zeros of characters of quasisimple groups is found in [19, 14, 17, 18, 12, 13, 15]. We shall use most of these results in this article. We need some definitions before we present the recent work of Lübeck and Malle [15]. Let Φn denote the n-th cyclotomic polynomial over ℚ. Let m,n be positive integers. Then, by m∥n, we mean that m∣n but m2∤n. If l>2 is not dividing q, the multiplicative order of q modulo l is denoted by dl⁢(q).

Theorem 2.1 ([15, Theorem 1]).

Let l>2 be a prime and M a finite quasisimple group of l-rank at least 3. Then, for any non-linear character χ∈Irr⁢(M), there exists an l-singular element g∈M with χ⁢(g)=0, unless either M is a finite group of Lie type in characteristic l, or l=5 and one of the following holds:

M=PSL5⁢(q) with 5∥(q-1), and χ is unipotent of degree χ⁢(1)=q2⁢Φ5,
M=PSU5⁢(q) with 5∥(q+1), and χ is unipotent of degree χ⁢(1)=q2⁢Φ10,
M=Ly and χ⁢(1)∈{48174,11834746},
M=E8⁢(q) with q odd, dl⁢(q)=4, and χ is one character in the Lusztig series of type D8.

Let ℳ be a simple, simply connected algebraic group over 𝔽¯p, the algebraic closure of a finite field of characteristic p, and let F:ℳ→ℳ be a Frobenius morphism such that M:=ℳF, the finite group of fixed points. Let ℳ* denote the dual group of ℳ with corresponding Frobenius morphism F*:ℳ*→ℳ*. Then M*:=(ℳ*)F* is the dual group of M. Using Deligne–Lusztig theory, we have that irreducible characters of M are partitioned into Lusztig series ℰ⁢(M,s*) that are parametrised by conjugacy classes of semisimple elements s* in the dual group M*. See [4, 6] for basic results on Deligne–Lusztig theory of complex representations of finite groups of Lie type.

The following lemma will be essential.

Lemma 2.2 ([10, Lemma 3.2]).

Let x∈M be semisimple, and let χ∈E⁢(M,s*) be an irreducible character of M with χ⁢(x)≠0. Then there is a maximal torus T≤M with x∈T such that T*≤CM*⁢(s*) for a torus T*≤M* which is a dual group of T.

For the rest of the section, we shall present some number theory results.

Lemma 2.3.

Let p be a prime and f a positive integer. Then the following statements hold:

If q=pf>11, then 6⁢f+1<(q2-q-2)/9.
If q=pf≥7 and q is odd, then 4⁢f+1<(q2-1)/8.

Let q,n≥2 be integers. Suppose that (q,n)≠(2,6), and if n=2 assume that q+1 is not a power of 2. Then, by Zsigmondy’s theorem [27], a Zsigmondy prime divisor l⁢(n) always exists. The Zsigmondy prime divisor is defined as a prime l⁢(n) such that

l(n)∣qn-1 but l(n)∤∏i=1n-1(qi-1).

Lemma 2.4.

Let q=pf for some prime p and a positive integer f. Suppose that a is a positive integer and b,c are non-negative integers.

If q-1=2c and q+1=2a⁢3b, then q=3,5 or 17.
If q-1=2a and q+1=2b⁢5c, then q=3 or 9.
If q-1=2a⁢5b and q+1=2c, then q=3.

Proof.

(a) If b=0, then q=3. Otherwise, we have

2=2a⁢(3b-2c-a),

so a=1 and 3b-1=2c-a. By Zsigmondy’s theorem, there is a Zsigmondy prime l∣3b-1 except when b≤2. If b=1, then q=5, and if b=2, then q=17.

(b) If c=0, then q=3. If c≥1, then a>b. Now

2=2b⁢5c-2a=2b⁢(5c-2a-b).

Since b≥1, we have that b=1 and 5c-2a-b=1, that is, 5c-1=2a-1. By Zsigmondy’s theorem [27], there exists a Zsigmondy prime l∣5c-1 unless when c=1. Hence q+1=10, and so q=9.

2=2c-2a⁢5b=2a⁢(2c-a-5b).

Hence a=1 and 2c-1-5b=1, which implies that 5b+1=2c-1. By [11, IX, Lemma 2.7], b=1, which can only happen when q=11. This is a contradiction since q+1=12≠2c. ∎

3 Quasisimple groups with a character vanishing on elements of the same order

In this section, we prove Theorem 1.1. In view of [16, Theorem 1.2], it is sufficient to only consider quasisimple groups M such that M/Z⁢(M) is isomorphic to a finite simple group of Lie type distinct from PSL2⁢(q).

Theorem 3.1.

Let M be a quasisimple group such that M/Z⁢(M) is a finite simple group of Lie type over a field of characteristic p distinct from PSL2⁢(q). If (*) holds for M, then M is one of the following:

M=PSU3⁢(4), χ⁢(1)=13,
M=B22⁢(8), χ⁢(1)=14.

3.1 Classical groups

We shall show that Theorem 3.1 (1) holds with a series of propositions.

We first show that the Steinberg character of a classical group of Lie type fails to satisfy (*).

Lemma 3.2.

Let M be a finite simple classical group of Lie type over a field of characteristic p, distinct from PSL2⁢(q). Then the Steinberg character χ of M fails to satisfy (*).

Proof.

Suppose that p=2. Then χ is of 2-defect zero, and so χ vanishes on every 2-singular element of M. In particular, χ vanishes on an involution. By [24, III, Theorem 5], M has an element of order 2⁢r for some odd prime r except when M≅PSL3⁢(4). The character table of PSL3⁢(4) exhibited in the Atlas [5] confirms our conclusion for this special case. We may assume that M has an element g of order 2⁢r with r as above. Then χ vanishes on g and so vanishes on two elements of distinct orders, contradicting (i) of (*).

Now we suppose that p is odd. Then χ is of p-defect zero, and so χ vanishes on every p-singular element of M. In particular, χ vanishes on a unipotent element of order p. Now M has an element g of order pr, where r is a prime number, since the size of the connected component of a prime graph of M containing p is at least 2 by [25, Theorem 1]. Hence χ⁢(g)=0, and the result follows. ∎

3.1.1 Special linear groups

Let ℳ=GLn⁢(𝔽¯p), and let F be the standard Frobenius map. The conjugacy classes of F-stable maximal tori of GLn⁢(𝔽¯p) and SLn⁢(𝔽¯p) are parametrised by conjugacy classes of Sn. Recall that conjugacy classes of Sn are parametrised by cycle shapes. If 𝒯≤GLn⁢(𝔽¯p) corresponds to λ=(λ1,λ2,…,λm)∈Sn such that λ1≥λ2≥…≥λm, then |T|=|𝒯F|=∏i=1m(qλi-1), and if 𝒯≤SLn⁢(𝔽¯p), then (q-1)⁢|T|=(q-1)⁢|𝒯F|=∏i=1m(qλi-1).

Lemma 3.3 ([12, Lemmas 3.1 and 4.1]).

Let λ⊢n be a partition and T a corresponding F-stable maximal torus of SLn⁢(F¯p) or SUn⁢(F¯p). Assume that either all parts of λ are distinct, or q≥3 and at most two parts of λ are equal. Then T=TF contains regular elements.

Lemma 3.4 ([12, Lemma 3.2]).

Let H≤PGLn⁢(F¯p) or H≤PGUn⁢(F¯p) be a reductive subgroup containing F-stable maximal tori corresponding to cycle shapes λ1,λ2,…,λr. If no intransitive or imprimitive subgroup of Sn contains elements of all these cycle shapes, then H=PGLn⁢(F¯p) or H=PGUn⁢(F¯p), respectively.

To use this result, we note that ℳ is connected reductive with a Steinberg endomorphism F:ℳ→ℳ and M:=ℳF. If 𝒯*≤𝐂ℳ*⁢(s*), then, since 𝒯* is connected, we have that 𝒯*≤𝐂ℳ*∘⁢(s*), a reductive subgroup of ℳ* (see [20, Theorem 14.2]).

Table 1 shows Zsigmondy primes li for the corresponding tori Ti. Note that elements of order li in the torus Ti are regular elements. It was shown in [18] that almost all characters of simple groups vanish on elements of order l1 or l2 whenever l1 and l2 exist.

Table 1

Tori and Zsigmondy primes for classical groups of Lie type.

M	\|T1\|	\|T2\|	l1	l2
An	(qn+1-1)/(q-1)	qn-1	l⁢(n+1)	l⁢(n)
An2 (n≥3 odd)	(qn+1-1)/(q+1)	qn+1	l⁢(n+1)	l⁢(2⁢n)
An2 (n≥2 even)	(qn+1+1)/(q+1)	qn-1	l⁢(2⁢n+2)	l⁢(n)
Bn,Cn (n≥3 odd)	qn+1	qn-1	l⁢(2⁢n)	l⁢(n)
Bn,Cn (n≥2 even)	qn+1	(qn-1+1)⁢(q+1)	l⁢(2⁢n)	l⁢(2⁢n-2)
Dn (n≥5 odd)	qn-1	(qn-1+1)⁢(q+1)	l⁢(n)	l⁢(2⁢n-2)
Dn (n≥4 even)	(qn-1-1)⁢(q-1)	(qn-1+1)⁢(q+1)	l⁢(n-1)	l⁢(2⁢n-2)
Dn2 (n≥4)	qn+1	(qn-1+1)⁢(q-1)	l⁢(2⁢n)	l⁢(2⁢n-2)

Since PSL3⁢(2)≅PSL2⁢(7) and PSL2⁢(7) is considered in [16, Theorem 1.2], we may assume that n=3 and q≥3 for the following result.

Proposition 3.5.

Let M be a quasisimple group such that M/Z⁢(M)=PSL3⁢(q), where q≥3. Then every non-trivial faithful irreducible character of M fails to satisfy (*).

Proof.

Using explicit character tables in the Atlas [5], we may assume that q≥13. First consider Z⁢(M)≠1. Now, |Z⁢(M)|=3, 3∣(q-1), and by (*), χ vanishes on a 3-element. Note that unipotent characters are not faithful when Z⁢(M)≠1. Hence we may assume that χ is not unipotent. Then χ lies in the Lusztig series ℰ⁢(M,s*) of a semisimple element s* in the dual group M*=PGL3⁢(q). Let T1 and T2 be tori of M corresponding to the partitions (3) and (2)⁢(1), respectively. By Lemma 3.3, the tori T1 and T2 contain regular elements. We claim that χ vanishes on regular elements either in T1 or in T2. Otherwise, by Lemma 2.2, 𝐂M*⁢(s*) contains conjugates of the duals T1* and T2*. This means that the corresponding reductive subgroup 𝐂ℳ*∘⁢(s*) contains 𝒯1* and 𝒯2*. Using Lemma 3.4, we have that 𝐂ℳ*∘⁢(s*)=PGL3⁢(𝔽¯p), that is, 𝐂M*⁢(s*)=PGL3⁢(q), and so χ is unipotent, contradicting our assumption that χ is not unipotent. Hence χ vanishes on regular elements in T1 or in T2. Suppose that χ vanishes on regular elements in T1. Note that |T1| is divisible by a Zsigmondy prime l1 and T1 contains regular elements of order l1. Since gcd⁡(l1,3)=1, χ vanishes on at least two elements of distinct orders, contradicting (*). We may thus assume χ vanishes on regular elements in T2. If q+1 is not a power of 2, then |T2| is divisible by a Zsigmondy prime l2. By the same argument as above, we may infer χ vanishes on at least two elements of distinct orders, contradicting (*). Suppose q+1 is a power of 2. This means that |T2| is even, and hence T2 contains elements of even order by [15, Remark 2.2]. Hence χ also vanishes on an element of even order, and the result follows.

Suppose M=PSL3⁢(q). Then χ is not the Steinberg character by Lemma 3.2. By [19, Theorem 2.1], χ vanishes on regular elements in T1 or in T2. Suppose that χ vanishes on regular elements of T1. Note that |T1| is divisible by a Zsigmondy prime l1. If |T1| is divisible by two distinct primes, then the result follows by [15, Remark 2.2]. Suppose that |T1| is a prime power. Then |T1|=q2+q+1gcd⁡(3,q-1) must be prime by [21]. Suppose

|T1|=q3-1(q-1)⁢gcd⁡(3,q-1)=q2+q+1gcd⁡(3,q-1)=l1.

Then G has

l1-13=q2+q-23⋅gcd⁡(3,q-1)

conjugacy classes whose elements are of order l1. Now

|Out⁢(M)|=2⋅gcd⁡(3,q-1)⋅f.

By Lemma 2.3, |Out⁢(M)|<6⁢f+1≤q2+q-29, and (ii) of (*) fails to hold. Suppose χ vanishes on regular elements in T2. By [19, Theorem 2.1], χ vanishes on elements of order q+1. If q is odd, then q+1 is even. In particular, q+1 is not a prime. By [18, Theorem 5.1], χ vanishes on an element of prime order which means that χ vanishes on two elements of distinct orders, contradicting (*). Hence we may assume that q is even so that q+1 is odd. We may assume that q+1 is prime by the above argument. Since |T2|=q2-1gcd⁡(3,q-1) and q-1gcd⁡(3,q-1)≠1, we have that |T2| is divisible by at least two primes. Hence there exists a prime l such that l∣(q-1) which entails the existence of an l-singular regular element in |T2| by [15, Remark 2.2]. By [19, Theorem 2.1], χ vanishes on this l-singular element. Hence χ vanishes on two elements of distinct orders, and the result follows. ∎

Proposition 3.6.

Suppose that M is quasisimple such that M/Z⁢(M)≅PSLn⁢(q), n≥4 and q≥2. Then every non-trivial faithful irreducible character of M fails to satisfy (*).

Proof.

Firstly, suppose that n≥4 and q=2. For M isomorphic to PSL4⁢(2) or PSL5⁢(2), we have explicit character tables in the Atlas [5], and for M/Z⁢(M) isomorphic to PSL6⁢(2) or PSL7⁢(2), we obtain explicit character tables in Magma [2]. Hence we may assume that n≥8. Then we have 3=q+1. Now (q+1)4∣|T| for a torus T corresponding to the partition (n-8)⁢(2)⁢(2)⁢(2)⁢(2). It follows that M is of 3-rank at least 4. Hence, by Theorem 2.1, χ vanishes on a 3-singular element. On the other hand, by [19, Theorem 2.1], if n is even, χ vanishes on an element of order qn/2+1 or an element of order qn-1-1, and if n is odd, then χ vanishes on an element of order qn-1 or an element of order q(n-1)/2+1. Note that, in all of the aforementioned cases, the order of elements on which χ vanishes, exceeds 3. Each such order is either relatively prime to 3 or is 3-singular. In the former case, χ vanishes on an element that is not of prime order. Using [18, Theorem 5.1], χ vanishes on an element of prime order. Hence χ vanishes on at least two elements of distinct orders, contradicting (*).

Suppose that M=SLn⁢(q), n≥4, q≥3, with Z⁢(M)≠1. By (*), |Z⁢(M)| is a power of a prime l that divides q-1, and χ necessarily vanishes on an l-element. We claim that χ also vanishes on an l1-element or an l2-element. Suppose the contrary. First note that χ is not a unipotent character since χ is faithful in M. Hence χ lies in the Lusztig series ℰ⁢(M,s*) of a semisimple element s* in the dual group M*=PGLn⁢(q). Let T1 and T2 denote maximal tori corresponding to the partitions (n) and (n-1)⁢(1). Note that T1 and T2 contain regular elements by Lemma 3.3. By Lemma 2.2, 𝐂M*⁢(s*) contains conjugates of the dual tori T1* and T2*. The corresponding reductive subgroup 𝐂ℳ*∘⁢(s*) contains 𝒯1* and 𝒯2*. Using Lemma 3.4, we infer that 𝐂ℳ*∘⁢(s*)=PGLn⁢(𝔽¯p), and so s* is central. Hence s*=1, and χ is unipotent, thus contradicting the assumption that χ is not unipotent. Hence our claim is true, and the result follows.

Suppose that M=PSLn⁢(q), n≥4, q≥3. First suppose that n=4 and q≥3. We have an explicit character table for PSL4⁢(3) in the Atlas [5], and for PSL4⁢(4) and PSL4⁢(5), we obtain an explicit character table in Magma. Assume that q≥7. Note that, for |T1| and |T2|, the Zsigmondy primes l1 and l2 exist, respectively. By the proof of [19, Theorem 2.1], χ is of l1-defect zero or l2-defect zero, and so χ vanishes on elements of order l1 or l2. Then

|T1|=q4-1(q-1)⁢gcd⁡(4,q-1)=(q+1)⁢(q2+1)gcd⁡(4,q-1)

is divisible by two distinct primes. Also,

|T2|=q3-1gcd⁡(4,q-1)=(q-1)⁢(q2+q+1)gcd⁡(4,q-1)

is divisible by two distinct primes since q-1gcd⁡(4,q-1)≠1. Hence χ vanishes on two regular elements of distinct orders.

Suppose n=5, q≥3. Assume that χ is not unipotent. Let T1, T2 and T3 be tori of M corresponding to the partitions (5), (4)⁢(1) and (3)⁢(2), respectively. These tori contain regular elements by Lemma 3.3. We claim that χ vanishes on regular elements in at least two of these tori. Otherwise, 𝐂M*⁢(s*) contains conjugates of the dual tori Ti* and Tj* of Ti and Tj, respectively, i≠j, 1≤i,j≤3, where χ lies in the Lusztig series ℰ⁢(M,s*). The corresponding reductive subgroup 𝐂ℳ*∘⁢(s*) contains 𝒯i* and 𝒯j*. It follows from Lemma 3.4 that 𝐂ℳ*∘⁢(s*)=PGL5⁢(𝔽¯q), that is, χ is unipotent, a contradiction. The claim is thus true. Now, for |T1| and |T2|, note that the corresponding Zsigmondy primes l1 and l2 exist, respectively. Hence χ vanishes on at least two elements of distinct orders l1, l2 or some positive integer that divides |T3|.

We may assume that χ is unipotent. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. It is sufficient to show that χ vanishes on an l-singular element with l≠5, an odd prime and gcd⁡(l1,l)=gcd⁡(l2,l)=1. Let q be even, and note that q≥3. If gcd⁡(5,q-1)=1, then there exists an odd prime l≠5 such that l divides (q-1) and M is of l-rank at least 3 and gcd⁡(l1,l)=gcd⁡(l2,l)=1. Hence χ vanishes on an l-singular element by Theorem 2.1. If gcd⁡(5,q-1)≠1, then there exists an odd prime l≠5 such that l∣(q+1), and so gcd⁡(l1,l)=gcd⁡(l2,l)=1. Note that M is of l-rank 2. Then, by the proof of [12, Proposition 3.8], χ vanishes on an l-singular element. Assume that q is odd. Suppose that gcd⁡(5,q-1)=1. Then there exists an odd prime l≠5 such that either l∣(q-1) or l divides (q+1), gcd⁡(l1,l)=gcd⁡(l2,l)=1 and M is of l-rank 2 with the following exception:

q-1=2a,a≥1 and q+1=2b⁢5c,b≥1,c≥0.

Then, by the proof of [12, Proposition 3.8], χ vanishes on an l-singular element for the former case. For the exceptions, q=3 or 9 by Lemma 2.4. If q=3, then using Magma [2] to calculate the character table of PSL5⁢(3), we conclude that χ does not satisfy (*). Let q=9. In this case, we look at the orders of T1 and T2. Now we have |T1|=95-19-1=112⋅61 and |T2|=94-1=25⋅5⋅41. Since χ is either of l1-defect zero or of l2-defect zero, χ vanishes on at least two elements of distinct orders. Assume that gcd⁡(5,q-1)=5. If there exists an odd prime l≠5 such that l∣(q-1) or l∣(q+1), then M is of l-rank at least 2 and by the proof of [12, Proposition 3.8], χ vanishes on an l-singular element. Hence the only exception we have is when q-1=2a⁢5b and q+1=2c. By Lemma 2.4, q=3, which does not satisfy gcd⁡(5,q-1)=5. Hence the result follows.

Suppose that n=6. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. If gcd⁡(6,q-1)=1, then there exists an odd prime l∣(q-1) such that the l-rank of M is 5. It follows that χ vanishes on at least two elements of distinct orders by Theorem 2.1 and since gcd⁡(l1,l)=gcd⁡(l2,l)=1. Let gcd⁡(6,q-1)=2. Then q is odd. If q≠3, then there exists an odd prime l such that l∣(q-1) or l∣(q+1). In this case, M is of l-rank at least 3, and we are done. If q=3, then using Magma [2] to calculate the character table of PSL6⁢(3), we conclude that χ does not satisfy (*). Let gcd⁡(6,q-1)=3 or 6. Then the 3-rank of M is 4, and the result follows.

Suppose that n=7. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. We first consider q even. If gcd⁡(7,q-1)=1, then there exists an odd prime l such that l∣(q-1) and the l-rank of M is 6. If gcd⁡(7,q-1)≠1, then since q is even, there exists an odd prime l≠7 such that l∣(q+1) and the l-rank of M is 3. Assume that q is odd. Suppose that gcd⁡(7,q-1)=1. Then there exists an odd prime l such that either l∣(q-1) or l∣(q+1) unless q=3. If q≠3, then we have an odd prime l and M is of l-rank at least 3. If q=3, then using Magma [2] to calculate the character table of PSL7⁢(3), we can conclude that χ fails to satisfy (*). Suppose gcd⁡(7,q-1)=7. If q+1 is not a power of 2, then there exists an odd prime l such that l∣(q+1), and we are done. We may thus assume that q+1=2a, a≥3. Now 3 divides either q-1, q or q+1. We know that 3∤(q+1). Suppose that 3∣(q-1). Then 3 is the desired odd prime. Thus 3∣q, that is, q=3f, f≥1. This implies that q=2a-1=3f. By [11, IX, Lemma 2.7], f=1, that is, q=3, a contradiction since gcd⁡(7,q-1)=7.

Suppose that n=8. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. If there exists an odd prime l such that l∣(q-1), then we are done. We may assume that q-1=2a, a≥1. Then q is odd. If there exists an odd prime l such that l∣(q+1), then we are done. Otherwise, q+1=2b, b≥2. Then q=3. For M=PSL8⁢(3), |T1| and |T2| are both divisible by two distinct primes. Since χ is of l1-defect zero or of l2-defect zero, we have that χ vanishes on two elements of distinct orders.

Suppose that n≥9. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. Consider a torus T of M corresponding to the partition (n-9)⁢(3)⁢(3)⁢(3). There exists a Zsigmondy prime l dividing q3-1 such that M is of l-rank at least 3. By Theorem 2.1, χ vanishes on an l-singular element. Since gcd⁡(l1,l)=gcd⁡(l2,l)=1, the result follows. This concludes our argument. ∎

3.1.2 Special unitary groups

Let ℳ=GLn⁢(𝔽¯p), and let F be the twisted Frobenius morphism. The conjugacy classes of F-stable maximal tori of GUn⁢(𝔽¯p) and SUn⁢(𝔽¯p) are also parametrised by conjugacy classes of Sn. If 𝒯≤GUn⁢(𝔽¯p) corresponds to the cycle shape λ=(λ1,λ2,…,λm)∈Sn with λ1≥λ2≥…≥λm, then

|T|=|𝒯F|=∏i=1m(qλi-(-1)λi)

whilst if 𝒯≤SUn⁢(𝔽¯p), then

(q+1)⁢|T|=(q+1)⁢|𝒯F|=∏i=1m(qλi-(-1)λi).

Proposition 3.7.

Let M be a quasisimple group such that M/Z⁢(M)=PSU3⁢(q), q≥3. If (*) holds for M, then M=PSU3⁢(4) with χ⁢(1)=13.

Proof.

We may conclude from the character tables in Atlas [5] that M=PSU3⁢(4) when 3≤q≤11. We may assume that q≥13. Note that χ is not the Steinberg character. We first consider the case M=SU3⁢(q) and Z⁢(M)≠1. Since we are only considering faithful characters, χ is not unipotent. Then |Z⁢(M)|=3, 3∣(q+1). By (iii) of (*), χ vanishes on a 3-element. We have that T1 and T2 correspond to the cycle shapes (3) and (2)⁢(1), and so T1 and T2 have regular elements by Lemma 3.3. Using the same argument as in Proposition 3.5, we have that χ vanishes on regular elements in T1 or in T2. If χ vanishes on regular elements in T1, then χ vanishes on an element of Zsigmondy prime order l1. Since gcd⁡(l1,3)=1, the result follows. If χ vanishes on regular elements in T2, then χ vanishes either on an element of Zsigmondy prime order l2 if q-1 is not a power of 2 or on an element of even order if q-1 is a power of 2. Since all the orders above are relatively prime to 3, χ vanishes on at least two elements of distinct orders, contradicting (*).

Let M=PSU3⁢(q). By [18, Lemmas 5.3 and 5.4], χ vanishes on regular elements in T1 or in T2. Assume that χ vanishes on regular elements in T1. Note that |T1| is divisible by a Zsigmondy prime l1. If |T1| is divisible by two distinct primes, then by [15, Remark 2.2], χ vanishes on at least two elements of distinct orders. Note that T1 is cyclic by [8, Section 3.3]. If |T1|=l1a, a>1, then χ vanishes on two elements of distinct orders l1a and l1, which contradicts (*). We may assume that

|T1|=q3+1(q+1)⁢gcd⁡(3,q+1)=q2-q+1gcd⁡(3,q+1)=l1.

If q=13, then M has 52 conjugacy classes of order 53, |Out⁢(M)|=2, contradicting (ii) of (*). We thus assume q≥16. Then M has

l1-13≥q2-q-23⋅gcd⁡(3,q+1)

conjugacy classes whose elements are of order l1. Now

|Out⁢(M)|≤2⋅gcd⁡(3,q-1)⋅f.

By Lemma 2.3, 6⁢f+1≤q2-q-29, and (ii) of (*) fails to hold.

We now consider the case where χ vanishes on regular elements in T2. By [19, Theorem 2.2], χ vanishes on an element of order q-1. On the other hand, χ vanishes on: an element of order Zsigmondy prime l2, an involution or on a regular unipotent element by the proof of [18, Lemma 5.4]. Therefore, χ vanishes on at least two elements of distinct orders, contradicting (*). ∎

Proposition 3.8.

Let M be a quasisimple group such that M/Z⁢(M)≅PSUn⁢(q), n≥4 and q≥2. Then every non-trivial faithful irreducible character of M fails to satisfy (*).

Proof.

We consider M/Z⁢(M)≅PSUn⁢(2) first. Using the character tables for PSU4⁢(2), PSU5⁢(2) and PSU6⁢(2) in Atlas [5], and for PSU7⁢(2), PSU8⁢(2) and PSU9⁢(2) from Magma [2], we may assume that n≥10. Suppose that Z⁢(M)≠1. This means that |Z⁢(M)|=3 and 3∣(q+1). Note that χ is not unipotent. By (*), χ vanishes on a 3-element. We claim that χ vanishes on regular elements in T1 or in T2. Assume that this claim is not true. Then 𝐂M*⁢(s*) contains conjugates of the dual tori T1* and T2* of T1 and T2 where χ lies in the Lusztig series ℰ⁢(M,s*). The corresponding reductive subgroup 𝐂ℳ*∘⁢(s*) contains the tori 𝒯1* and 𝒯2*. By Lemma 3.4, 𝐂ℳ*∘⁢(s*)=PGLn⁢(𝔽¯2) and s* is central. Hence we have 𝐂M*⁢(s*)=PGUn⁢(2), and so χ is unipotent, a contradiction. The claim is true, and χ vanishes either on an l1-element or on an l2-element. Hence χ vanishes on at least two elements of distinct orders.

Assume that Z⁢(M)=1. By [19, Theorem 2.2], we have that χ vanishes on elements of order l1 or l2. Consider a torus T of M corresponding to the partition (n-10)⁢(2)⁢(2)⁢(2)⁢(2)⁢(2). Hence M is of l-rank at least 3, where l=q+1=3. By Theorem 2.1, χ vanishes on an l-singular element. Hence the result follows.

Suppose that M/Z⁢(M)≅PSUn⁢(q), n≥4, q≥3. Assume that Z⁢(M)≠1. By (*), |Z⁢(M)| is a power of a prime l∣(q+1), and χ vanishes on an l-element. Using the proof of [19, Theorem 2.2], χ vanishes on an l1-element or an l2-element, and the result follows.

Suppose M≅PSUn⁢(q). By the proof of [19, Theorem 2.2], χ is of l1-defect zero or l2-defect zero. Suppose n≥9, and consider a torus T of M corresponding to the partition (n-9)⁢(3)⁢(3)⁢(3). Then there exists a Zsigmondy prime l=l⁢(6) dividing q3+1, and M is of l-rank at least 3. By Theorem 2.1, χ vanishes on an l-singular element, and the result follows. Hence we may assume that n≤8.

Suppose that n=8. Recall that q≥3. If gcd⁡(8,q+1)=1, then there exists an odd prime l∣(q+1) such that the l-rank of M is 7. By Theorem 2.1, χ vanishes on an l-singular element, and the result follows since gcd⁡(l1,l)=gcd⁡(l2,l)=1. If gcd⁡(8,q+1)≠1, then q is odd and there exists an odd prime l such that l∣(q-1) or l∣(q+1) unless q=3. If q≠3, then M is of l-rank at least 3, and hence χ vanishes on an l-singular element for an odd prime l by Theorem 2.1. If q=3, then using Magma [2] to calculate the character table of PSU8⁢(3), we can conclude that χ fails to satisfy (*). Hence the result follows.

Suppose that n=7. We first consider the case when q is even. If we have gcd⁡(7,q+1)=1, then there exists an odd prime l≠7 such that l∣(q+1). If gcd⁡(7,q+1)≠1, then since q is even, there exists an odd prime l≠7 such that l∣(q-1). In both cases, M is of l-rank at least 3, and so χ vanishes on an l-singular element. Since gcd⁡(l1,l)=gcd⁡(l2,l)=1, χ vanishes on at least two elements of distinct orders. Assume that q is odd. Suppose that gcd⁡(7,q+1)=1. Then there exists an odd prime l≠7 such that l∣(q-1) or l∣(q+1) unless q=3. If q≠3, then we have an odd prime l and M is of l-rank at least 3, which means that χ vanishes on at least two elements of distinct orders. If q=3, then using Magma [2] to calculate the character table of PSU7⁢(3), we conclude that χ does not satisfy (*). Suppose gcd⁡(7,q+1)=7. If q-1 is not a power of 2, then there exists an odd prime l≠7 such that l∣(q-1). Hence M is of l-rank at least 3, and gcd⁡(l1,l)=gcd⁡(l1,l)=1. Thus χ vanishes on two elements of distinct orders, a contradiction to (*).

We may assume that q-1=2a, a≥3. Now 3 divides either q-1, q or q+1. We know that 3∤(q-1). Suppose that 3∣(q+1). Then 3 is the desired odd prime since M is of 3-rank at least 3. Thus 3∣q, that is, q=3f, f≥1. This implies that q-1=3f-1=2a. By Zsigmondy’s theorem, there is a Zsigmondy prime l∣(3f-1) unless f≤2. If f=1, then we have q=3, contradicting the hypothesis that gcd⁡(7,q+1)=7. If f=2, then q=9, again contradicting the hypothesis that gcd⁡(7,q+1)=7.

Suppose n=6. If gcd⁡(6,q+1)=1, then there exists an odd prime l∣q+1 such that the l-rank of M is 5. Let gcd⁡(6,q+1)=2. Then q is odd. If q≠3, then there exists an odd prime l such that l∣(q-1) or l∣(q+1), and the result follows since M is of l-rank at least 3. Let gcd⁡(6,q+1)=3. Then q is even. Note that q≥4. Then there exists an odd prime l that divides q-1. Hence χ vanishes on an l-singular element since the l-rank of M is 6. Let gcd⁡(6,q+1)=6. Then q is odd. If there exists an odd prime l≠3 such that l∣(q+1), then the result follows. We may assume that q+1=2a⁢3b, a≥1 and b≥1. Then there exists an odd prime l≠3 that divides q-1, and the result follows unless q-1=2c, c≥3. Hence we may assume that q-1=2c. By Lemma 2.4, q=5 or 17 since gcd⁡(6,q+1)=6. In both cases, the 3-rank of M is 5 by Theorem 2.1. Hence the result follows.

Suppose n=5. Assume that χ is not unipotent. Let T1, T2 and T3 be tori of M corresponding to (5), (4)⁢(1) and (3)⁢(2), respectively. These tori contain regular elements by Lemma 3.3. We claim that χ vanishes on regular elements in at least two of these tori. Otherwise, 𝐂M*⁢(s*) contains conjugates of the dual tori Ti* and Tj* of Ti and Tj, respectively, i≠j, 1≤i,j≤3, where χ lies in the Lusztig series ℰ⁢(M,s*). The corresponding reductive subgroup 𝐂ℳ*∘⁢(s*) contains 𝒯i* and 𝒯j*. It follows from Lemma 3.4 that 𝐂ℳ*∘⁢(s*)=PGU5⁢(𝔽¯q), that is, χ is unipotent, a contradiction. The claim is thus true. Now, for |T1| and |T2|, note that the corresponding Zsigmondy primes l1 and l2 exist, respectively. Hence χ vanishes on at least two elements of distinct orders l1, l2 or some positive integer that divides |T3|.

Assume that χ is unipotent. Then χ vanishes on elements of order l1 or l2 by the proof of [19, Theorem 2.1]. By Theorem 2.1, it is sufficient to show that χ vanishes on an l-singular element with l≠5, an odd prime. Let q be even, and note that q≥3. If gcd⁡(5,q+1)=1, then there exists an odd prime l≠5 such that l∣(q+1) and M is of l-rank at least 3. If gcd⁡(5,q+1)≠1, then there exists an odd prime l≠5 such that l∣(q-1). Note that M is of l-rank 2. By the proof of [12, Proposition 4.2], χ vanishes on an l-singular element. Now assume that q is odd. Suppose that gcd⁡(5,q+1)=1. Then there exists an odd prime l≠5 such that l∣(q-1) or l∣(q+1) with the following exception: q-1=2a, a≥1 and q+1=2b⁢5c, b≥1, c≥0. By Lemma 2.4, q=3 or 9. In both cases, using Magma [2] to calculate the character tables of PSU5⁢(3) and PSU5⁢(9), we conclude that χ does not satisfy (*). Assume that gcd⁡(5,q+1)=5. If there exists an odd prime l≠5 such that l∣(q-1) or l∣(q+1), then M is of l-rank at least 2, and we are done by [12, Proposition 4.2]. Hence the only exception we have is when q-1=2a⁢5b and q+1=2c. Lemma 2.4 entails q=3 which contradicts the assumption that gcd⁡(5,q+1)=5.

First suppose that n=4 and q≥3. We have an explicit character table for PSU4⁢(3) in the Atlas [5], and for PSU4⁢(4) and PSU4⁢(5), we obtain an explicit character table in Magma. Assume that q≥7. Note that, for |T1| and |T2|, the Zsigmondy primes l1 and l2 exist, respectively. By the proof of [19, Theorem 2.1], χ is of l1-defect zero or l2-defect zero, and so χ vanishes on elements of order l1 or l2. Then

|T1|=q4-1(q+1)⁢gcd⁡(4,q+1)=(q-1)⁢(q2+1)gcd⁡(4,q+1)

is divisible by two distinct primes. Also,

|T2|=q3+1gcd⁡(4,q+1)=(q+1)⁢(q2-q+1)gcd⁡(4,q+1)

is divisible by two distinct primes since q+1gcd⁡(4,q+1)≠1. Hence χ vanishes on two regular elements of distinct orders. ∎

3.1.3 Symplectic groups and special orthogonal groups

Let ℳ be a simple, simply connected algebraic group of type Bn, Cn or Dn over 𝔽¯p, and let F:ℳ→ℳ be a Frobenius morphism such that M:=ℳF. Then the ℳF-conjugacy classes of F-stable maximal tori of ℳ are parametrised by the conjugacy classes of W, the Weyl group of ℳ. If ℳ is of type Bn or Cn, then W is isomorphic to the wreath product C2≀Sn, and the conjugacy classes of W are parametrised by pairs of partitions (λ,μ)⊢n (see [15, Section 2.1] for details). In particular, if a maximal torus T=𝒯F corresponds to a partition (λ,μ)=((λ1,λ2,…,λr),(μ1,μ2,…,μs))⊢n, then

|T|=∏i=1r(qλi-1)⁢∏j=1s(qμj+1),

and 𝒯F contains cyclic subgroups of orders qλi-1 and qμj+1 for all i and j.

If ℳ is of type Dn, then W=C2n-1⋊Sn, and the ℳF-conjugacy classes of F-stable maximal tori of ℳ are parametrised by pairs of partitions (λ,μ)⊢n such that μ has an even number of parts if ℳF is Spin2⁢n+⁢(q) and μ has an odd number of parts if ℳF is the non-split orthogonal group Spin2⁢n-⁢(q). Now |T| is the same as in the case when ℳ is of type Bn or Cn, that is,

|T|=∏i=1r(qλi-1)⁢∏j=1s(qμj+1).

Lemma 3.9 ([15, Lemma 2.1]).

Let M be a simple, simply connected classical group of type Bn, Cn or Dn defined over F¯p with corresponding Steinberg morphism F.

Let (λ,μ)=((λ1,λ2,…,λr),(μ1,μ2,…,μs)) be a pair of partitions of n, and T a corresponding F-stable maximal torus of M. Then T=TF contains regular elements if one of the following is fulfilled:

q>3, λ1<λ2<⋯<λr and μ1<μ2<⋯<μs;
q∈{2,3}, λ1<λ2<⋯<λr, μ1<μ2<⋯<μs, all λi≠2, and if ℳ is of type Bn or Cn, then also all λi≠1;
ℳ is of type Dn, 2<λ1<λ2<⋯<λr, 1=μ1=μ2<μ3<⋯<μs.

Lemma 3.10 ([15, Lemma 2.3]).

Let M be a simple algebraic group of type Bn, Cn(with n≥2) or Dn (with n≥4) with Frobenius endomorphism F such that MF is a classical group. Let Λ be a set of pairs of partitions (λ,μ)⊢n. Assume the following:

there is no 1≤k≤n-1 such that all (λ,μ)∈Λ are of the form
(λ1,μ1)⊔(λ2,μ2) 𝑤𝑖𝑡ℎ⁢(λ1,μ1)⊢k;
the greatest common divisor of all parts of all (λ,μ)⊢n is 1 ;
if ℳ is of type Bn, then there exist pairs (λ,μ)⊢n for which μ has an odd number of parts, and one for which μ has an even number of parts.

If s∈MF is semisimple such that CM⁢(s) contains maximal tori of M corresponding to all (λ,μ)∈Λ, then s is central.

We first consider a quasisimple group M such that M/Z⁢(M)≅PSp4⁢(q). Since Sp4⁢(2)′≅PSL2⁢(9), PSp4⁢(3)≅PSU4⁢(2), and the groups PSL2⁢(9) and PSU4⁢(2) were dealt with in [16, Theorem 1.2] and Proposition 3.8, respectively, we shall not consider them in the result below.

Proposition 3.11.

Let M be a quasisimple group such that M/Z⁢(M)≅PSp4⁢(q), where q≥4. Then every non-trivial faithful irreducible character of M fails to satisfy (*)

Proof.

Since the character tables Sp4⁢(4) and Sp4⁢(5) are in Atlas [5], we may assume that q≥7. Suppose that q is even. Then the result follows from the generic character tables in Chevie [9]. We may assume that q is odd, q≥7. For this case, we first suppose Z⁢(M)≠1. Note that χ is not unipotent. Then |Z⁢(M)|=2, and by (*), χ vanishes on a 2-element. For each prime l such that l∣(q-1) or l∣(q+1), the Sylow l-subgroups of G are non-cyclic. Since q≠3, there exists an odd prime l such that l∣(q-1) or l∣(q+1). By [15, Theorem 4.1], χ vanishes on an l-singular element. But gcd⁡(2,l)=1, so χ vanishes on at least two elements of distinct orders, as required.

Suppose M≅PSp4⁢(q). We may assume that χ is not the Steinberg character. By the proof of [19, Theorem 2.3], χ vanishes on regular elements in T1 or in T2. In particular, we may choose two conjugacy classes 𝒞1 and 𝒞2 in T1 and T2 such that χ vanishes on 𝒞1 or 𝒞2. Now 𝒞2 may contain elements which are not of Zsigmondy prime order. In that case, the result follows since χ vanishes on elements of prime order by [18, Theorem 5.1]. Hence we may assume that 𝒞1 and 𝒞2 contain elements of Zsigmondy prime orders. Suppose that χ vanishes on elements in T2. Note that |T2|=q2-12 is even. Hence T2 contains a regular element of even order by [15, Remark 2.2], and χ vanishes on this element. This means that χ vanishes on two elements of distinct orders, contradicting (*). Assume that χ vanishes on elements of T1. Note that T1 is cyclic by [8, Section 4.5]. If |T1| is not prime, then there exist at least two elements of distinct orders on which χ vanishes. We may assume that |T1|=q2+12 is prime. Then there are

(q2+1)2-14=q2-18

conjugacy classes with elements of order q2+12. On the other hand, we have that |Out(M/Z(M)|≤4f, where q=pf, p is a prime and f≥1. By Lemma 2.3, 4⁢f+1<q2-18, and the result follows by (ii) of (*). ∎

Let 𝒮={PSp2⁢n⁢(q)∣n≥3}∪{PSO2⁢n+1⁢(q)∣n≥3}∪{PSO2⁢n±⁢(q)∣n≥4}.

Proposition 3.12.

Let M be a quasisimple group such that M/Z⁢(M)∈S. Then every non-linear faithful irreducible character χ of M fails to satisfy (*).

Proof.

Note that χ is not the Steinberg character by Lemma 3.2. We first consider the case where M∈𝒮 with q=2. Since we have character tables in Atlas [5] for Sp2⁢n⁢(2)≅SO2⁢n+1⁢(2), 3≤n≤4, and PSO2⁢n±⁢(2), 4≤n≤5, we may assume that n≥5 and n≥6, respectively. Since q+1=3, then M is of 3-rank at least 5. By Theorem 2.1, χ vanishes on a 3-singular element. For M≅Sp2⁢n⁢(2), χ vanishes on elements of order l1 or elements of order l2 in Table 1, or χ is of l3-defect zero, where l3=l⁢(n-1) (the last case only arising when n is even) by [18, Lemmas 5.3–5.5]. Note that Zsigmondy primes l1, l2, l3 exist, and we have gcd⁡(l1,3)=gcd⁡(l2,3)=gcd⁡(l3,3)=1. Hence χ vanishes on at least two elements of distinct orders, contradicting (*). For M≅PSO2⁢n±⁢(2), n≥6, χ vanishes on elements of order l1, or l2 or χ is of l3-defect zero, where l3=l⁢(2⁢n-4) (the last case only arising when n is even) by [18, Lemmas 5.3, 5.4 and 5.6]. Since the Zsigmondy primes l1, l2 and l3 exist, the result follows.

Henceforth, we may assume that q≥3 and n≥3. Suppose that Z⁢(M)≠1. Then gcd⁡(2,q-1)=2, and by (*), χ vanishes on a 2-element. We want to show that χ also vanishes on an element of Zsigmondy prime order. Note that χ is not unipotent, and so χ lies in the Lusztig series ℰ⁢(M,s*) of s* in the dual M*. Let ((λ),(μ)) and ((λ′),(μ′))) be the partitions corresponding to tori T1 and T2 with orders in Table 1. These tori contain regular elements by Lemma 3.9. We claim that χ vanishes on regular elements in at least one of these tori. Otherwise, by Lemma 2.2, 𝐂M*⁢(s*) contains conjugates of the dual tori T1* and T2*. The corresponding subgroup 𝐂ℳ*⁢(s*) contains conjugates of the dual tori 𝒯1* and 𝒯2*. It follows from Lemma 3.10 that s* is central. Hence 𝐂M*⁢(s*)=M*, that is, χ is unipotent, a contradiction. The claim is thus true. Now, for T1 and T2, note that the Zsigmondy primes l1 and l2 exist in respect of |T1| and |T2|. Hence χ vanishes on at least two elements of distinct orders, and we are done.

Suppose that Z⁢(M)=1. Consider M≅PSp2⁢n⁢(q), n≥3, or PSO2⁢n+1⁢(q), n≥3. By [18, Lemmas 5.3–5.5], χ vanishes on elements of order l1, l2, or χ is of l3-defect zero, where l3=l⁢(n-1) (the last case arising when n is even). In all cases, the Zsigmondy primes exist. Now there exists an odd prime l such that l∣(q-1) or l∣(q+1) except when q=3. Note that M is of l-rank at least 3. If q≠3, then by Theorem 2.1, χ vanishes on an l-singular element. Since we have gcd⁡(l1,l)=gcd⁡(l2,l)=gcd⁡(l3,l)=1, the result follows. We are left with the case when q=3. If n≥6, then M has a torus T corresponding to (-,(n-6)⁢(2)⁢(2)⁢(2)), i.e., M is of l-rank at least 3, where l∣(q2+1). The result follows again. Hence we may assume that n≤5, that is,

M∈{PSp6⁢(3),PSp8⁢(3),PSp10⁢(3),PSO7⁢(3),PSO9⁢(3),PSO11⁢(3)}.

We have explicit character tables for PSp6⁢(3) and PSO7⁢(3) in the Atlas [5], and using Magma [2] for the rest of the groups, we have our conclusion.

Suppose that Z⁢(M)=1 and M≅PSO2⁢n-⁢(q) with n≥4 and q≥3. By the proof of [19, Theorem 2.5], χ is of l1-defect zero or of l2-defect zero. If q≠3, then there exists an odd prime l such that l∣(q-1) or l∣(q+1) and M is of l-rank at least 3. By Theorem 2.1 and since gcd⁡(l1,l)=gcd⁡(l2,l)=1, the result then follows. Consider n≥4 and q=3. We have an explicit character table for PSO8-⁢(3) in the Atlas [5], and for PSO10-⁢(3), we obtain an explicit character table in Magma. For n=6, the orders of T1 and T2 are divisible by two distinct primes, and the result follows. We may assume that n≥7. Hence M is of l-rank at least 3 when l∣q2+1. Therefore, by Theorem 2.1, χ vanishes on at least two elements of distinct orders.

Suppose that Z⁢(M)=1 and M≅PSO2⁢n+⁢(q) with n≥4 and q≥3. Assume that n is odd. Then the Zsigmondy primes l1 and l2 exist, and χ vanishes on regular elements in T1 or in T2 by [18, Lemma 5.3]. If q≠3, then there exists an odd prime l such that l∣(q-1) or l∣(q+1) and M is of l-rank at least 3. Hence χ vanishes on an l-singular element by Theorem 2.1, and thus χ vanishes on at least two elements of distinct orders. Let q=3. If n≥7, then consider a torus corresponding to the cycle shape (-,(n-6)⁢(2)⁢(2)⁢(2)). It follows that M is of l-rank at least 3 with l∣(q2+1), and by Theorem 2.1, χ vanishes on an l-singular element. Hence we may assume n≤5. Hence n=5, and so M≅PSO10+⁢(3). Using Magma [2], the result follows.

Suppose that n≥4 is even. By [18, Lemma 5.6], χ vanishes on regular elements of order l1 or l2 or χ is of l3-defect zero, where l3=l⁢(2⁢n-4). An argument similar to that used above allows us to dispose of the case when q≠3. Suppose q=3. Now if n≥8, then M is of l-rank at least 3, where l is an odd prime dividing q2+1. In particular, l=5. By Theorem 2.1, χ vanishes on a 5-singular element. Since gcd⁡(l1,5)=gcd⁡(l2,5)=gcd⁡(l3,5)=1, the result follows. If n=4, that is, M≅PSO8+⁢(3), then we have the explicit character table in the Atlas [5], and if n=6, we obtain an explicit character table in Magma for PSO12+⁢(3). This concludes our proof. ∎

3.2 Exceptional groups

3.2.1 Exceptional groups of small Lie rank

Since PSL2⁢(8)≅G22⁢(3)′ and PSU3⁢(3)≅G2⁢(2)′, and PSL2⁢(8), PSU3⁢(3) were dealt with in [16, Theorem 1.2] and Proposition 3.7, respectively, we exclude them in this section.

Let

ℒ={B22⁢(q2)∣q2=22⁢f+1,f≥1}∪{G22⁢(q2)∣q2=32⁢f+1,f≥1}∪{F42⁢(q2)⁢∣q2>⁢2}∪{G2⁢(q)∣q≥3}∪{D43⁢(q)∣q≥2}.

Proposition 3.13.

Let M be a quasisimple group such that

M/Z⁢(M)∈ℒ∪{F42⁢(2)′}.

If (*) holds for M, then M=B22⁢(8) with χ⁢(1)=14.

Proof.

The simple group M=B22⁢(8) satisfies the conclusion of our proposition from its character table in the Atlas [5]. For the rest of the groups, using explicit character tables in Atlas [5] and generic ordinary character tables in Chevie [9], we may conclude that every non-trivial character of M does not satisfy conditions of (*). Hence the result follows. ∎

3.2.2 Exceptional groups of large Lie rank

Table 2 shows the Zsigmondy primes li for the corresponding tori Ti. It was shown in [18] that every non-trivial irreducible character which is not the Steinberg character vanishes on an element of order li for some i=1,2,3.

Table 2

Tori and Zsigmondy primes for groups of Lie type.

M	\|T1\|	\|T2\|	\|T3\|	l1	l2	l3
F4⁢(q)	Φ12	Φ8		l⁢(12)	l⁢(8)
E6⁢(q)	Φ12⁢Φ3	Φ9	Φ8⁢Φ2⁢Φ1	l⁢(12)	l⁢(9)	l⁢(8)
E62⁢(q)	Φ18	Φ12⁢Φ6	Φ8⁢Φ2⁢Φ1	l⁢(18)	l⁢(12)	l⁢(8)
E7⁢(q)	Φ18⁢Φ2	Φ14⁢Φ2	Φ12⁢Φ3⁢Φ1	l⁢(18)	l⁢(14)	l⁢(12)
E8⁢(q)	Φ30	Φ24	Φ20	l⁢(30)	l⁢(24)	l⁢(20)

Proposition 3.14.

Let M be a quasisimple exceptional finite group of Lie type over a field of characteristic p and of rank at least 4. Then every non-trivial faithful irreducible character of M fails to satisfy (*).

Proof.

Note that the group M must be one of these types: F4, E6, E62, E7 or E8.

Suppose that M/Z⁢(M)≅F4⁢(2). Using the character tables in the Atlas [5], the result follows. Let M=F4⁢(q), q≥3. Then M is simple. From [18, Lemma 5.9], we have that χ vanishes on regular elements of order l1, l2 or l3=l⁢(3), or χ is of p-defect zero if χ is the Steinberg character. On the other hand, there exists an odd prime l such that l∣(q-1) or l∣(q+1) and M is of l-rank at least 3 unless q=3. If q≠3, then by Theorem 2.1, we have that χ vanishes on an l-singular element, and since gcd⁡(l1,l)=gcd⁡(l2,l)=gcd⁡(l3,l)=gcd⁡(p,l)=1, the result follows. If q=3, then using an explicit character table of M=F4⁢(3) from Magma [2], the result follows.

Now suppose M=E6⁢(q), q≥2, with Z⁢(M)=1. From [18, Lemma 5.9], we have that χ vanishes on regular elements of order l1, l2 or l3, or χ is of p-defect zero if χ is the Steinberg character. On the other hand, M is of l-rank at least 3 for an odd prime l such that l∣(q3-1). By Theorem 2.1, χ vanishes on an l-singular element. Since gcd⁡(l1,l)=gcd⁡(l2,l)=gcd⁡(l3,l)=gcd⁡(p,l)=1, we are done. Now suppose that Z⁢(M)≠1, that is, |Z⁢(M)|=3. By (*), χ vanishes on a 3-element. Using the above argument, χ vanishes on an l-singular element, where l≠3 is an odd prime such that l∣(q2+q+1). Since gcd⁡(3,l)=1, the result follows.

Suppose that M=E62⁢(q), q≥2, with Z⁢(M)=1. By [18, Lemma 5.9], we have that χ vanishes on regular elements of order l1, l2 or l3, or χ is of p-defect zero if χ is the Steinberg character. On the other hand, M is of l-rank at least 3 for an odd prime l≠3 such that l∣(q2-q+1). By Theorem 2.1, χ vanishes on an l-singular element. Since gcd⁡(l1,1)=gcd⁡(l2,l)=gcd⁡(l3,l)=gcd⁡(p,l)=1, the result follows. Now suppose that Z⁢(M)≠1, so |Z⁢(M)|=3 and q=3⁢b-1 for some positive integer b≥2 using the Atlas [5] for E62⁢(2). We may assume that q≥5. It is sufficient to show that M is of l-rank at least 3 for some prime l≠3. The candidate for l is an odd prime l such that l∣(q2-q+1). Finally, let M/Z⁢(M)=E62⁢(2) with |Z⁢(M)|=2 since, by (*)(iii), |Z⁢(M)| is a prime power. Then the character table in the Atlas [5] concludes this case.

We will use the same arguments for M=E7⁢(q) and M=E8⁢(q) in the case when M is simple. By [18, Lemma 5.9], we have that χ vanishes either on regular elements of order l1, l2 or l3, or χ is of p-defect zero if χ is the Steinberg character. By Theorem 2.1, χ vanishes on an l-singular element, where l is an odd prime such that l∣(q2-q+1). Hence the result follows.

Now suppose M=E7⁢(q) and Z⁢(M)≠1. By (*), χ vanishes on a 2-element. Using Theorem 2.1, χ vanishes on an l-singular element, where l is an odd prime such that l∣(q3+1). Hence the result follows. ∎

4 Non-solvable groups with a character vanishing on one class

4.1 Almost simple groups of Lie type

We are in a position to prove Theorem 1.2.

Proof of Theorem 1.2.

Suppose that χ∈Irr⁢(G) is faithful, primitive and vanishes on exactly one conjugacy class. By [16, Theorem 3.3], there exist normal subgroups M and Z of G such that G/Z is an almost simple group and M is a quasisimple group. In particular, it was shown in [16] that the normal subgroup M necessarily satisfies (*). For M/Z isomorphic to a sporadic simple group, alternating group An, n≥5, or PSL2⁢(q), q≥4, the result follows from [16, Theorems 1.2 and 1.3]. For M/Z isomorphic to a finite group of Lie type distinct from PSL2⁢(q), q≥4, the result follows from our results in Section 3.

Conversely, suppose that one of (1)–(7) in the hypothesis holds. For (1)–(6), the result follows from [16, Theorem 1.3]. Now, when G=B22⁢(8):3 with χ⁢(1)=14, by the same argument above [16, Theorem 5.2], the primitivity of χ follows. ∎

Communicated by Timothy C. Burness

Funding statement: The author acknowledges the support of DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Opinions expressed and conclusions arrived at are those of the author and should not necessarily to be attributed to the CoE-MaSS.

Acknowledgements

This work will form part of the author’s PhD thesis, and he would like to thank his advisors, Professors Tong-Viet and van den Berg, and Dr. Le, for their helpful suggestions. He would also like to thank Professor Malle for the careful reading of an earlier version of this article and pointing out some gaps in the proofs.

References

[1] C. Bessenrodt and J. R. B. Olsson, Weights of partitions and character zeros, Electron. J. Combin. 11 (2004/06), no. 2, Research Paper 5. 10.37236/1862Search in Google Scholar

[2] W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. 10.1006/jsco.1996.0125Search in Google Scholar

[3] T. C. Burness and H. P. Tong-Viet, Derangements in primitive permutation groups, with an application to character theory, Q. J. Math. 66 (2015), no. 1, 63–96. 10.1093/qmath/hau020Search in Google Scholar

[4] R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1985. Search in Google Scholar

[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University, Eynsham, 1985. Search in Google Scholar

[6] F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Math. Soc. Stud. Texts 21, Cambridge University, Cambridge, 1991. 10.1017/CBO9781139172417Search in Google Scholar

[7] J. D. Dixon and A. Rahnamai Barghi, Irreducible characters which are zero on only one conjugacy class, Proc. Amer. Math. Soc. 135 (2007), no. 1, 41–45. 10.1090/S0002-9939-06-08628-XSearch in Google Scholar

[8] P. C. Gager, Maximal tori in finite groups of Lie type, PhD thesis, University of Warwick, 1973. Search in Google Scholar

[9] M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput., 7 (1996), no. 3, 175–210. 10.1007/BF01190329Search in Google Scholar

[10] R. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces, J. Amer. Math. Soc. 25 (2012), no. 1, 77–121. 10.1090/S0894-0347-2011-00709-1Search in Google Scholar

[11] B. Huppert and N. Blackburn, Finite Groups. II, Grundlehren Math. Wiss. 242, Springer, Berlin-New York, 1982. 10.1007/978-3-642-67994-0Search in Google Scholar

[12] C. Lassueur and G. Malle, Simple endotrivial modules for linear, unitary and exceptional groups, Math. Z. 280 (2015), no. 3–4, 1047–1074. 10.1007/s00209-015-1465-0Search in Google Scholar

[13] C. Lassueur, G. Malle and E. Schulte, Simple endotrivial modules for quasi-simple groups, J. Reine Angew. Math. 712 (2016), 141–174. 10.1515/crelle-2013-0100Search in Google Scholar

[14] F. Lübeck and G. Malle, (2,3)-generation of exceptional groups, J. Lond. Math. Soc. (2) 59 (1999), no. 1, 109–122. 10.1112/S002461079800670XSearch in Google Scholar

[15] F. Lübeck and G. Malle, A Murnaghan–Nakayama rule for values of unipotent characters in classical groups, Represent. Theory 20 (2016), 139–161. 10.1090/ert/480Search in Google Scholar

[16] S. Y. Madanha, On a question of Dixon and Rahnamai Barghi, Comm. Algebra 47 (2019), no. 8, 3064–3075. 10.1080/00927872.2018.1549667Search in Google Scholar

[17] G. Malle, Almost irreducible tensor squares, Comm. Algebra 27 (1999), no. 3, 1033–1051. 10.1080/00927879908826479Search in Google Scholar

[18] G. Malle, G. Navarro and J. R. B. Olsson, Zeros of characters of finite groups, J. Group Theory 3 (2000), no. 4, 353–368. 10.1515/jgth.2000.028Search in Google Scholar

[19] G. Malle, J. Saxl and T. Weigel, Generation of classical groups, Geom. Dedicata 49 (1994), no. 1, 85–116. 10.1007/BF01263536Search in Google Scholar

[20] G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Stud. Adv. Math. 133, Cambridge University, Cambridge, 2011. 10.1017/CBO9780511994777Search in Google Scholar

[21] T. Nagel, Des équations indéterminées x2+x+1=yn et x2+x+1=3⁢yn, Norsk. Mat. Foren. Skr. Serie I 2 (1921), 12–14. Search in Google Scholar

[22] G. Navarro, Irreducible restriction and zeros of characters, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1643–1645. 10.1090/S0002-9939-00-05747-6Search in Google Scholar

[23] G. Qian, Finite solvable groups with an irreducible character vanishing on just one class of elements, Comm. Algebra 35 (2007), no. 7, 2235–2240. 10.1080/00927870701302222Search in Google Scholar

[24] M. Suzuki, Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. 99 (1961), 425–470. 10.1090/S0002-9947-1961-0131459-5Search in Google Scholar

[25] J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487–513. 10.1090/pspum/037/604579Search in Google Scholar

[26] E. Žmud’, Finite groups having an irreducible complex character with a class of zeros, Dokl. Akad. Nauk SSSR 247 (1979), no. 4, 788–791. Search in Google Scholar

[27] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284. 10.1007/BF01692444Search in Google Scholar

Received: 2019-04-20

Revised: 2019-08-06

Published Online: 2019-12-04

Published in Print: 2020-03-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2019-2051