Zeros and roots of unity in character tables

For any finite group $G$, Thompson proved that, for each $\chi\in {\rm Irr}(G)$, $\chi(g)$ is a root of unity or zero for more than a third of the elements $g\in G$, and Gallagher proved that, for each larger than average class $g^G$, $\chi(g)$ is a root of unity or zero for more than a third of the irreducible characters $\chi\in {\rm Irr}(G)$. We show that in many cases"more than a third"can be replaced by"more than half".

The author suspects that the answers to these questions are both 1/2.In particular, we propose the following: Conjecture 1. θ(G) and θ ′ (G) are ≥ 1/2 for every finite group G.
We establish the conjecture for all finite nilpotent groups by establishing a much stronger result about zeros for this family of groups, which includes all p-groups.The number of p-groups of order p n was shown by G. Higman [6] and C. C. Sims [12] to equal p 2 27 n 3 +O(n 8/3 ) with n → ∞, and it is a folklore conjecture that almost all finite groups are nilpotent in the sense that the number of nilpotent groups of order at most n the number of groups of order at most n which, in view of our result, would mean that Conjecture 1 holds for almost all finite groups.Conjecture 1 is readily verified for rational groups, such as Weyl groups, and all groups of order < 2 9 , and although θ(G) = 1/2 for certain dihedral groups, the second inequality is strict in all known cases.The author suspects that both inequalities are strict for all finite simple groups: Conjecture 2. θ(G) and θ ′ (G) are > 1/2 for every finite simple group G.

Nilpotent groups
We begin with our results on finite nilpotent groups.
Theorem 2. Let G be a finite nilpotent group, and let g ∈ G.
for at least half of the nonlinear χ ∈ Irr(G).
The key ingredient in the proof of Theorems 1 and 2 is Theorem 8, which will replace the result of Siegel used by Thompson and Gallagher.Its proof relies on some auxiliary results of independent interest and is based on arithmetic in cyclotomic fields.
For each positive integer k, we denote by ζ k a primitive k-th root of unity.For any algebraic integer α contained in some cyclotomic field, we denote by l(α) the least integer l such that α is a sum of l roots of unity, by f(α) the least positive integer k such that α ∈ Q(ζ k ), and by m(α) the normalized trace Proof of Lemma 4. If n = 0, then there is nothing to prove, so assume n ≥ 1.Let ζ be a primitive p n -th root of unity.For each α j and β k , let r j and s k be nonnegative integers such that α j = ζ r j and β k = ζ s k .Put Then P (ζ) = 0, so P (x) is divisible in Z[x] by the cyclotomic polynomial Φ p n (x) = Φ p (x p n−1 ).
Proposition 5. Let G be a finite group, let χ ∈ Irr(G), and let g be an element of G with order a power of a prime p.If p = 2 or χ(1) ≡ ±2 (mod p), then either χ(g) = 0, χ(g) is a root of unity, or m(χ(g)) ≥ 2.
Proof of Proposition 5. Suppose that p = 2 or χ(1) ≡ ±2 (mod p).Let p n be the order of g, and let ζ be a primitive p n -th root of unity, so (1) We will show that either α = 0, α is a root of unity, or m(α) ≥ 2.
), then P divides p n .If P = 1, then α is rational and the conclusion follows.If P is divisible by p 2 , then for γ a primitive P -th root of unity, α is uniquely of the shape the α k are algebraic integers, and a straightforward calculation [2, p. 115] shows that m(α) is at least the number of nonzero α k .By (1), at least two of the α k are nonzero.Hence m(α) ≥ 2 if p 2 | P .It remains to consider the case Lemma 3].So assume l(α) = 2. Then by [9, Thm.1(i)], α can be written in the shape By Lemma 4, By ( 3) and the fact that χ(1) ≡ ±2 (mod p), Hence, for some root of unity ρ and primitive p-th root of unity ξ, α = (ξ − 1)ρ. Hence Lemma 6.Let G be a finite group, let χ ∈ Irr(G), and let g be an element of G with order a power of a prime p.If χ(1) ≡ ±1 (mod p), then χ(g) is not a root of unity.
Proof of Lemma 6.Let p n be the order of g, so χ(g) ∈ Q(ζ p n ), and suppose that χ(g) is a root of unity.Since the roots of unity in a given cyclotomic field Q(ζ k ) are the l-th roots of unity for l the least common multiple of 2 and k, we then have for some ǫ ∈ {1, −1} and p n -th root of unity ξ.So by Lemma 4, either χ(1) ≡ 1 (mod p) or χ(1) ≡ −1 (mod p).
Lemma 7. Let G be a finite group of prime-power order, let g ∈ G, and let Proof of Lemma 7. If |G| = p n with p prime, then each g ∈ G has order a power of p, and each χ ∈ Irr(G) has degree a power of p.So if χ(1) > 1, then by Proposition 5 and Lemma 6, for each g ∈ G, χ(g) = 0 or m(χ(g)) ≥ 2.
For any character χ of a finite group, let Theorem 8. Let G be a finite nilpotent group, let χ ∈ Irr(G), and let g ∈ G. Then Proof of Theorem 8.
Since G is nilpotent, it is the direct product of its nontrivial Sylow subgroups P 1 , P 2 , . . ., P n .Let g 1 , g 2 , . . ., g n be the unique sequence with g k ∈ P k and g = g 1 g 2 . . .g n .
Proof of Theorem 1.By Proposition 9.
Proof of Theorem 2. Taking the relation applying the elements σ of the Galois group G = Gal(Q(ζ |G| )/Q), and averaging over G, we have So for L = {χ ∈ Irr(G) : By Theorem 8, for each χ ∈ N , From ( 16) and ( 17 Proof of Corollary 3. By Theorem 1 and Theorem 2.

Simple groups
We now establish Conjecture 2 for several families of simple groups.
Maintaining the notation of Suzuki [13], there are elements σ, ρ, ξ 0 , ξ 1 , ξ 2 such that each element of G can be conjugated into exactly one of the sets where A i = ξ i (i = 1, 2, 3), and the irreducible characters of G are given by the following table [13,Theorem 13]: and ǫ j 1 and ǫ k 2 are certain characters on A 1 and A 2 .The A i 's satisfy and denoting by G i (i = 0, 1, 2) the set of elements g ∈ G that can be conjugated into A i − {1}, we have where l 0 = 2 and √ −1/(q−1) and s ∈ Z. Then and γ s = 0 ⇔ 4s ± (q − 1) ≡ 0 (mod 2(q − 1)).
By ( 29) and ( 26)-( 27), Equality must hold in (31) because By (30) and the fact that, for any Equality must hold in (32) because 1 G , σ G , and ρ G have size < |G|/|Cl(G)|, and for any g Verification of III.Let q = p n with p prime, G = L 2 (q), let R and S be as in [7, pp. 402-403], and let G 0 (resp.G 1 ) be the set of nonidentity elements g ∈ G that can be conjugated into R (resp.S ).
The irreducible characters of G are given by Ward [14] in a 16-by-16 table, with the last 6 rows being occupied by 6 families of exceptional characters, the sizes of which are, from top to bottom, q − 3 4 , q − 3 4 , q − 3 24 , q − 3 8 , q − 3m 6 , q + 3m 6 .
From Ward's table, we find that for any class g G ∈ {1 G , X G , J G }, χ(g) ∈ {0, 1, −1} for more than half of the irreducible characters χ of G. Since the classes 1 G , X G , J G all have size < |G|/|Cl(G)|, we conclude that θ ′ (G) > 1/2.The first step in verifying θ(G) > 1/2 is to compute the following table: Then with Table 1 and Ward's table in hand, a straightforward check establishes that, for each χ ∈ Irr(G), Verification of V and VI.Here, in Tables 2 and 3, we report the values of θ and θ ′ for sporadic groups and simple groups of order ≤ 10 9 .All values are rounded to the number of digits shown.

Table 2 .
The sporadic groups.