A Lower Bound on the Number of Generators of a Defect Group

Assuming that the statement of the Alperin–McKay–Navarro conjecture holds for a p-block B with defect group D, we show that the number of generators of D is bounded from below by the number of height-zero characters in B fixed by a specific element of the absolute Galois group of the rational numbers.


Introduction
One of the main themes in the Representation Theory of Finite Groups is the study of how the values of characters in a p-block B of a finite group G and the structure of a defect group of B influence each other, where p is a prime. The aim of this note is to investigate connections between the values of height-zero characters in a p-block B and generation properties of the defect groups of B.
Given a finite p-group D, recall that the size of a minimal set of generators of D is m if, and only if, |D : Φ(D)| = p m , where Φ(D) is the Frattini subgroup of D. In this note we refer to the number m as the rank of D.
If B is a p-block of a finite group G with defect group D then the height-zero characters of B, here denoted by Irr 0 (B), are those χ ∈ Irr(B) such that χ(1) p = |G : D| p , where n p is the maximal power of p dividing the integer n. The group Gal(Q/Q) acts naturally on the set Irr(G) preserving degrees by χ σ (g) = σ (χ(g)) for every g ∈ G and χ ∈ Irr(G). Define σ ∈ Gal(Q/Q) as the element that sends p-power roots of unity to their p + 1 power and fixes roots of unity of order prime to p. Then the action of σ permutes the elements of Irr 0 (B) for every block B. In the following, we will denote by Irr 0,σ (B) the set of fixed-points of Irr 0 (B) under the action of σ and by k 0,σ (B) the size of Irr 0,σ (B).
Recent advances on the subject [14,18] suggest the existence of a mutual influence between k 0,σ (B) and the rank of D (at least for primes p ≤ 3). We propose the following: The above conjectural bound is related to Brauer's k(B)-conjecture [1, Problem 20] and Olsson's conjecture [16]. The former predicts that k(B) ≤ |D|, where k(B) = |Irr(B)| is the number of irreducible characters in the block B, and the latter that k 0 (B) ≤ |D : D |, where k 0 (B) = |Irr 0 (B)|. Notice that while Irr 0 (B) is always non-empty, the existence of some σ -fixed character in Irr 0 (B) is not trivially guaranteed unless D = 1. In Theorem 2.3 we show that Conjecture (1) holds for any p-block B with normal defect group. In particular, any p-block B satisfying the statement of the so-called Alperin-McKay-Navarro conjecture [13, Conjecture B] also satisfies the statement of Conjecture (1). Since the Alperin-McKay-Navarro conjecture is known to hold for p-solvable, sporadic, symmetric and alternating groups by [4,5,13,20] and for all p-blocks with cyclic defect groups by [13,Theorem 3.4], then Theorem 2.3 implies that Conjecture (1) holds for all p-blocks of these families of groups and for all p-blocks with cyclic defect groups.
A natural problem in this context is to study under which conditions is the upper bound in Conjecture (1) attained. Recall that a character χ ∈ Irr(G) is p-rational if all the values of χ lie in the cyclotomic extension Q(ξ ), where ξ is a root of unity of order |G| p . Here n p is the p -part of any natural number n so that n = n p n p . By the definition of σ , every p-rational character of G is σ -fixed.
In the case where p = 2, the relationship between k 0,σ (B) and |D : Φ(D)| seems closer.  [14], we believe that the following can be true: Unfortunately, the statements in Conjectures (2) and (3) do not seem to generalize to higher rank defect groups. If N is an elementary 2-group of rank 4 and H ∈ Syl 5 (Aut(H )), then the group G = N H has a unique 2-block B, namely the principal one, and k 0, Conjectures (2) and (3)  We care to mention that, for p odd, the σ -fixed characters of G are the almost prational characters of G defined in [6]. (We note that the definition of almost p-rational characters generalizes at the same time the classical notion of p-rationality and the notion of almost rationality defined in [8].) For p = 2, almost 2-rational characters are just 2-rational characters. In this case, it can be shown that the σ -fixed irreducible characters of G of odd degree are 2-rational (see Theorem 2.5).

On Conjectures (1) and (3)
For a fixed prime p, consider the set Bl(G) of Brauer p-blocks of G as in [12], so that Bl(G) induces a partition of Irr(G). We write Irr(B) to denote the subset of characters in Irr(G) that belong to B for B ∈ Bl(G). Every block B has associated a uniquely defined conjugacy class of p-subgroups of G, namely its defect groups. Given a block B of G with defect group D, we write B ∈ Bl(G|D) and we let b ∈ Bl(N G (D)|D) denote its Brauer first main correspondent. We write Irr 0 (B) to denote the subset of height-zero characters in Irr(B), as in Section 1.
The group G = Gal(Q(e 2πi/|G| )/Q) acts on Irr(G) permuting the subsets {Irr 0 (B) | B ∈ Bl(G)} by [12,Theorem 3.19]. Define H ≤ G to be the subgroup consisting of those elements τ ∈ G for which there exists some fixed f ∈ N such that τ (ξ) = ξ p f for every root of unity of order prime to p in Q(e 2πi/|G| ). We write Irr 0,τ (B) to denote the subset of τ -fixed characters in Irr 0 (B). Notice that we can see the Galois automorphism σ defined in Section 1 as an element of H by restricting it to Q(e 2πi/|G| ). Actually σ acts on each Irr 0 (B) for B ∈ Bl(G) (meaning that σ permutes trivially the set {Irr 0 (B) | B ∈ Bl(G)}) because σ fixes every Brauer character of G.

The Alperin-McKay-Navarro conjecture
We will see that the statements of Conjectures (1) and (3) follow from the statement of the Alperin-McKay-Navarro conjecture together with important results which already appeared in the literature. In order to do so, we recall some results on the theory of blocks with a normal defect group. We follow the notation of [12,Chapter 9]. Let B ∈ Bl(G|D) and assume that D G. Write C = C G (D). We will denote by b ∈ Bl(CD|D) a root of B, and we will let θ ∈ Irr(b) be the canonical character associated with B (see [12,Theorem 9.12] and the subsequent discussion). Recall that D ⊆ ker(θ) and θ has p-defect zero when viewed as a character of CD/D (that is, θ(1) p = |CD : D| p ). In this situation, the set Irr(b) is parametrized by the set Irr(D). We may write Irr(b) = {θ λ | λ ∈ Irr(D)}, where θ λ is defined as in [12,Theorem 9.12] and ker(λ) ⊆ ker(θ λ ). Then

Irr(G|θ λ ).
It is not difficult to see that height-zero characters of B lie over characters θ λ parametrized by linear characters λ of D, so that Irr 0 (B) =

Irr(G|θ λ ) ⊆ Irr(G/D ).
If N G, we will often identify with Irr(G/N ) the irreducible characters χ ∈ Irr(G) with N ⊆ ker(χ ). In the above expression we are using that kind of identifications (as for Lin(D) = Irr (D/D )).
An explicit description of the set Irr 0,σ (B) when the defect group of B is normal was given in [18]. This is [18, Lemma 1.2] which we restate below. G be a finite group and let p be a prime. Suppose that B is a block of G  with a normal defect group D. Let b be a root of B with canonical character θ. Then  [12, pp.198-199] for further details). The following key lemma was shown in [10].

Lemma 2.2 Let B be a p-block of a finite group G with defect group D. Suppose that D G, then B dominates a unique blockB of G/Φ(D). In particular,
Proof The first part follows by [10,Corollary 4]. In the following, for any τ ∈ G we denote by Irr 2 ,τ (G) the set of odd-degree irreducible characters of G fixed under the action of τ . Theorem 2.5 Let G be a finite group and let τ ∈ Gal(Q(e 2πi/|G| )/Q(e 2πi/|G| 2 )) be such that the subfield of Q(e 2πi/|G| 2 ) fixed by τ is Q( √ ε2), with ε ∈ {±1}. Then the set Irr 2 ,τ (G) consists of 2-rational characters.
Let |G| = 2 n m with m odd. Let Q m = Q(e 2πi/m ). By restriction, we can see σ as an element of Gal(Q(e 2πi/|G| )/Q m ) ∼ = Gal(Q(e πi/2 n−1 )/Q) of order 2 n−2 . Moreover, a character χ is σ -fixed if, and only if, Q m (χ ) = Q m (χ (g) | g ∈ G) ⊆ Q m ( √ 2i). By Theorem 2.5, the odd-degree irreducible characters fixed under the action of σ are 2rational. In particular, for blocks B of maximal defect, the invariant k 0,σ (B) counts the number of 2-rational characters of odd-degree in B.