Counting positive defect irreducible characters of a finite group

Laurence Barker, Bilkent University.

Abstract: Let z_{+}(G) be the number of ordinary irreducible characters of a finite group G which have positive defect with respect to a prime p. We express z_{+}(G) as the p-adic limit of a sequence of enumerative parameters of G and p. When p = 2, and under a suitable hypothesis on the Sylow 2-subgroups of G, we give two local characterisations of the parity of z_{+}(G), one of them compatible with Alperin’s Weight Conjecture, the other apparently independent.

1: A formula for the number of positive defect irreducible characters

Richard Brauer’s theory of finite group representations is an analysis of the representations of a finite group G (usually variable) in terms of a prime p (always fixed). Analysing the theory itself, three strands emerge. Firstly, — to borrow a phrase of Brauer — we may consider “arithmetic properties” of (ordinary absolutely) irreducible characters. For instance, can we express conveniently the number z_{+}(G) of positive defect irreducible characters? (The defect of an irreducible character χ of G is the non-negative integer d such that χ(1)_{p} p^{d} = |G|_{p}.) Secondly, we may consider the group algebra RG where R is a P-adic completion of some ring of algebraic integers, and P is a prime divisor of (p). For instance, it is well-known that if the field of fractions of R splits for G, then z_{+}(G) is determined by the isomorphism class of the centre ZRG. Thirdly, we may consider representations of “local” subgroups (by which we roughly mean non-trivial p-subgroups, their normalisers, and let us say, central extensions of factor groups of such normalisers.) For instance, Alperin’s Weight Conjecture [1] asserts that z_{+}(G) is determined by the “local” subgroups of G. A precise definition of the term “locally determined” is suggested in Thévenaz [18].

Since rings of algebraic integers, when they arise as ground rings, are often usefully replaced with P-adic completions, might not even enumerative parameters of “arithmetic properties” of irreducible characters be usefully regarded as p-adic integers? But the entwinement of the first two strands is easy; the miracle of Brauer’s theory (so it seems to me) is in the braiding of the third with the other two. We shall think of z_{+}(G) as a p-adic integer, and see if this helps us obtain “local” information about it.

Another, entirely different angle on the material below is to deliberately ignore most of the content of Michel Broué’s memorable remark (from a seminar in 1991) “Alperin’s Conjecture must be the combinatorial shadow of some deeper algebraic structure”. While considerable effort is being focused on searches for such “deeper algebraic structure” (see, for a small sample, Dade [8], Ellers [9], Robinson [16]), the bald fact is that Alperin’s Conjecture is a “combinatorial” assertion. It is even enumerative.

We shall end this section with Theorem 1.1, a formula expressing z_{+}(G) as the p-adic limit of a sequence of enumerative parameters of G. To discuss Sections 2 and 3, let us asume that p = 2 and that the Sylow 2-subgroups of G are abelian. (Actually, our hypothesis on the Sylow 2-subgroups will be weaker, but for now let us minimise technicalities.) Section 2 may be seen mainly as an application of Theorem 1.1 to prove that Alperin’s Conjecture holds up to parity. In Section 3, Theorem 1.1 is used to deduce Theorem 3.4, a local characterisation of the parity of z_{+}(G). My failure to derive Theorem 3.4 from Alperin’s Conjecture is, to me, more interesting than success would have been. Speculation on improving Theorem 3.4 is left largely to the reader, but in connection with this, in Section 4, we point out a relationship between two formulas expressing z_{0}(G) as the p-rank of an integer matrix.

Let K be an algebraically closed field of characteristic zero. The number k(G) of conjugacy classes of G, and the number z_{0}(G) of defect-zero irreducible characters of G are related by

k(G) = z_{0}(G) + z_{+}(G).

Given a positive integer n, let us write ω(n, G)|G| for the number of (2 + n)-tuples (x, y, g_{1}, ...g_{n}) consisting of elements of G such that each g_{i} is a p-element, not all the g_{i} are trivial, and [x, y] = g_{1}...g_{n}. (By definition, [x, y] := xyx^{−1}y^{−1}.)

The essential origin of the following result is Strunkov [17]. We have also drawn an idea from Iizuka–Watanabe [12, page 58].

Theorem 1.1: As a congruence of p-local integers,

−z_{+}(G) ≡_{p^{n}} ω(n, G)

for all positive integers n. In particular, taking the p-adic limit,

−z_{+}(G) = lim_{n→∞} ω(n, G).

Proof: For each irreducible character χ of G, let e_{χ} denote the primitive idempotent of ZKG such that χ(e_{χ}) = χ(1). Let G_{p}^{+} denote the sum in KG of the p-elements of G. Let ϕ be the class function on G such that ϕ(g) is the number of pairs (x, y) of elements of G such that [x, y] = g. We have

(G_{p}^{+})^{n} = ∑_{χ}(χ(G_{p}^{+})/χ(1))^{n}e_{χ}

where χ runs over the irreducible characters of G. By calculating, in two different ways, the ordinary trace of each class sum acting by translation on ZKG — firstly with respect to the basis consisting of the class sums, and secondly with respect to the basis consisting of the idempotents e_{χ} — we deduce that

ψ = |G| ∑_{χ} χ/χ(1).

A theorem of Frobenius in Curtis–Reiner [7, 41.11] implies that each χ(G_{p}^{+}) is divisible by |G|_{p}. If χ has defect zero, then χ(G_{p}^{+}) = χ(1). Therefore, |G|_{p} divides ϕ((G_{p}^{+})^{n}), and

z_{0}(G) ≡_{p^{n}} ϕ((G_{p}^{+})^{n})/|G|.

We complete the argument by observing that

ω(n, G) = ϕ((G_{p}^{+})^{n} − 1)/|G| = ϕ((G_{p}^{+})^{n})/|G| − k(G).

⊓ ⊔

2: Application to Alperin’s Conjecture

We apply Theorem 1.1 to Alperin’s Conjecture using techniques from Knörr– Robinson [13], and earlier related works by S. Bouc, K. S. Brown, D. Quillen, Thévenaz, and Webb. First, we need some notation, and must briefly discuss some results in [13].

Given a poset P, we write sd(P) for the set of chains \underline{x} = (x_{0}, ...x_{n}) in P, partially ordered by the subchain relation. (We disallow the empty chain.) The length of such a chain \underline{x} is defined to be n(\underline{x}) = n. The simplicial complex with vertex set P and simplex set sd(P) will also be denoted as P. (The rationale for our notation is that, as simplicial complexes, sd(P) is the barycentric subdivision of P.) The Euler characteristic of P is

χ(P) = ∑_{\underline{x}∈sd(P)}(−1)^{n(\underline{x})}.

For any G-set X, and x ∈ X, we write N_{G}(x) for the stabiliser of x in G. When the notation x ∈_{G} X is used to index the terms of a sum, x is understood to run over representatives of the G-orbits of X.

Recall that a G-poset is a poset upon which G acts as automorphisms. We write S(G) for the G-poset consisting of the non-trivial p-subgroups of G, partially ordered by inclusion, and with G acting by conjugation. We write A(G) for the G-subposet of S(G) consisting of the non-trivial elementary abelian p-subgroups. A non-trivial p-subgroup P of G is said to be radical in G provided P = O_{p}(N_{G}(P)); then we call N_{G}(P) a radical normaliser in G. We say that a chain (P_{0} < ... < P_{n}) in sd(S(G)) is radical in G provided each P_{i} is radical in N_{G}(P_{0} < ... < P_{i−1}) (interpreting the grotesque expression N_{G}(P_{0} < ... < P_{−1}) to mean G). As a weaker condition, we say that (P_{0} < ... < P_{n}) is normal provided each P_{i} is normal in P_{n}. Thus

rad(S(G)) ⊆ nor(S(G)) ⊆ sd(S(G))

where the elements of rad(S(G)) and nor(S(G)) are, respectively, the radical and normal chains. The G-sets rad(S(G)) and nor(S(G)) were introduced in Knörr– Robinson [13].

Proposition 2.1: (Knörr–Robinson): Let f be a function defined on the set of subgroups of G, invariant on each conjugacy class of subgroups, and taking values in some abelian group. Then the value of

∑_{\underline{P} ∈_{G}X}(−1)^{n(\underline{P})}f(N_{G}(\underline{P}))

is the same for all four choices of

X ∈ {sd(A(G)), sd(S(G)), nor(S(G)), rad(S(G))}.

Proposition 2.2 is given in Knörr–Robinson [13]. We must comment briefly on their proof because we shall need to adapt part of it below. Chain-pairing arguments in [13, 3.3] show that the three sums with X ∈ {sd(A(G)), sd(S(G)), nor(S(G))} have the same value. As noted in [13, page 52], another chain-pairing argument shows that the two sums with X ∈ {nor(S(G)), rad(S(G))} have the same value. (Details are spelt out in [2, 2.2].) Part of Proposition 2.2 was also discovered by Webb, and is illuminated in Thévenaz–Webb [19, Theorem 2].

Let ℓ(G) denote the number of irreducible p-modular characters of G.

Theorem 2.2: (Knörr–Robinson) Let X be a family of isomorphism classes of finite groups which is closed under radical normaliser subgroups. If any one of the following equalities always holds whenever G belongs to X, then all three do.

ℓ(G) = z_{0}(G) + ∑_{P ∈_{G}S(G)} z_{0}(N_{G}(P)/P),

z_{0}(G) = ℓ(G) − ∑_{\underline{P} ∈_{G}sd(S(G))}(−1)^{n(\underline{P})}ℓ(N_{G}(\underline{P})),

z_{+}(G) = ∑_{\underline{P} ∈_{G}sd(S(G))}(−1)^{n(\underline{P})}k(N_{G}(\underline{P})).

Furthermore, for any positive integer m, the assertion still holds when the equality relation = is replaced by the congruence relation ≡_{m}.

Theorem 2.2 is a slight generalisation of Knörr–Robinson [13, 3.8, 4.5]. The demonstration in [13] extends appropriately with little change. Alternatively, arguing as in [2, 3.2], Theorem 2.2 follows quickly from [13, 4.5] and [2, 3.1]. (To dispell any bemusement concerning the hypothesis on X, it is worth pointing out that, in the first of the three sums, if z_{0}(N_{G}(P)/P) is non-zero, then P is radical in G, whereupon z_{0}(N_{G}(P)/P) is determined by the radical normaliser N_{G}(P).)

Henceforth, S will always denote a Sylow p-subgroup of G. The proof of the following remark is routine.

Remark 2.3: Given a chain \underline{P} ∈ nor(S(G)), and writing m_{G}(\underline{P}) for the number of G-conjugates of \underline{P} belonging to nor(S(S)), then as p-local integers,

m_{G}(\underline{P})/|G:N_{G}(\underline{P})| ≡_{p} 1.

Let ω(G) := ω(1, G), and let Ω(G) be the set of pairs (x, y) of elements of G such that [x, y] is a non-trivial p-element. Thus |G|ω(G) = |Ω(G)|. We seek some “local” way of counting the elements of Ω(G) (up to congruence modulo p|G|_{p}).

Let Γ(S, G) be the subset of Ω(G) consisting of the pairs (x, y) such that no non-trivial subgroup of S is normalised by both x and y. Let

Δ(S, G) := Ω(G) − Γ(S, G) = ⋃_{P ∈S(S)} Ω(N_{G}(P)).

We define γ(G) := |Γ(S, G)|/|G| and δ(G) := |Δ(S, G)|/|G|. Then

ω(G) = γ(G) + δ(G).

Note that γ(G) and δ(G) are independent of the choice of S. Since S acts freely on Γ(S, G), the rational numbers γ(G) and δ(G) are p-local integers.

Theorem 2.4: As p-local integers,

z_{+}(G) + γ(G) ≡_{p} ∑_{\underline{P} ∈sd(S(G))}(−1)^{n(\underline{P})}k(N_{G}(\underline{P})).

Proof: By Theorem 1.1, the assertion is equivalent to the congruence

δ(G) ≡_{p} − ∑_{\underline{P} ∈sd(S(G))}(−1)^{n(\underline{P})}k(N_{G}(\underline{P})).

Given (x, y) ∈ Δ(S, G), let S(x, y) denote the unique largest subgroup of S normalised by x and y. The construction of Δ(S, G) ensures that S(x, y) is non-trivial. We have

\{\underline{P} ∈ sd(S(S)):\{x, y\} ⊆ N_{G}(\underline{P})\} = sd(S(S(x, y))).

Also, S(S(x, y)) has a unique maximal element, so as a simplicial complex, S(S(x, y)) is contractible. Therefore

δ(G) = ∑_{(x,y)∈Δ(S,G)} χ(S(S(x, y)))/|G| = ∑_{\underline{P} ∈sd(S(S))}(−1)^{n(\underline{P})}ω(N_{G}(\underline{P}))|N_{G}(\underline{P})|.

Adaptations of chain-pairing arguments used to prove Proposition 2.1 show that the indexing set sd(S(S)) may be replaced with sd(S(S)) ∩ nor(S(G)). Therefore,

δ(G) = ∑_{\underline{P} ∈_{G}nor(S(G))}(−1)^{n(\underline{P})}ω(N_{G}(\underline{P})) |N_{G}(\underline{P})| m_{G}(\underline{P}).

By Theorem 1.1, each ω(N_{G}(\underline{P})) ≡_{p} −z_{+}(N_{G}(\underline{P})) = −k(N_{G}(\underline{P})). Proposition 2.1 and Remark 2.3 now finish the proof. ⊓ ⊔ Adapting another chain-pairing argument used to prove Proposition 2.1 gives

δ(G) = ∑_{\underline{P} ∈_{G}rad(S(G))}(−1)^{n(\underline{P})}ω(N_{G}(\underline{P})) |N_{G}(\underline{P})| m_{G}(\underline{P}).

For each radical chain \underline{P}, the term ω(N_{G}(\underline{P}))|N_{G}(\underline{P})|m_{G}(\underline{P}) depends only on N_{G}(\underline{P}) and G. It can thence be shown that δ(G) depends only on the Lefschetz invariant of S(G), and so depends only on the G-homotopy class of S(G); see Benson [4, Chapter 6] for the terminology and techniques.

Concerning the final result in this section, let us note that a sufficient condition for a finite 2-group P to have a unique maximal elementary abelian subgroup is that P is a direct product of abelian, generalised quaternion, and modular groups Mod_{2^{n}} with n ≠ 3. Further examples can be constructed using semidirect products.

Theorem 2.5: Suppose that p = 2, and that a Sylow 2-subgroup of G has a unique maximal elementary abelian subgroup. Then the three equalities in Theorem 2.2 all hold up to parity.

Proof: Let Γ := Γ(S, G). By Theorem 2.4, it suffices to show that 2|S| divides |Γ|. Let α and β be involutions in D_{8} such that αβ has order 4, and let D_{8} act on Γ such that α(x, y) = (y, x) and β(x, y) = (x, y^{−1}) for all (x, y) ∈ Γ. Since the conjugation action of S on Γ is free, we may assume, for a contradiction, that some element (x, y) ∈ Γ lies in an S × D_{8}-orbit of order |S|. By the hypothesis on G, the stabiliser of (x, y) in S ×D_{8} must intersect non-trivially with 1×D_{8}. But x ∉ {y, y^{−1}} because [x, y] ≠ 1, hence (x, y) ∈ {(x^{−1}, y), (x, y^{−1})}. Also, each image of (x, y) under D_{8} is S-conjugate to (x, y). Therefore x and y are both involutions. Writing 2^{n} for the order of the element (xy)^{2} = [x, y], then x and y generate a dihedral group of order 2^{n+2}. This contradicts the hypothesis on G. ⊓ ⊔

3: Möbius inversion

The proof in Knörr–Robinson [13] of Theorem 2.1 is an inversion argument based on the G-poset S(G). Some of the proof is recast in [2] as a Möbius inversion argument on a directed multigraph. However, it is not a Möbius inversion argument on S(G). Here, we combine Theorem 1.1 with a Möbius inversion argument on S(G), obtaining local information about z_{+}(G) which appears to be independent of the results and conjectures in [13]. The following explicit formula for Möbius inversion in a family of p-groups is an immediate consequence of Kratzer–Thévenaz [14, 2.4]; see also Hall [10, 1.4], [11, Section 3].

Theorem 3.1: (Kratzer–Thévenaz): Let P be a family of p-groups such that all chains (of inclusions) in P are of finite length, and whenever P ≤ R ≤ Q with P, Q ∈ P, then R ∈ P. Let α and β be two functions on P taking values in some abelian group.

(1) The following two identities, taken over all P ∈ P, are equivalent:

α(P) = ∑_{Q∈P : Q≤P} β(Q),

β(P) = ∑_{r≥0}(−1)^{r}p^{r(r−1)/2} ∑_{Q∈P : P/Q≅C_{p}^{r}} α(Q),

where the notation Q ∈ P : P/Q ≅ C_{p}^{r} indicates that Q runs over the normal subgroups of P in P such that P/Q is elementary abelian of rank r.

(2) The following two identities, for P ∈ P, are equivalent:

α(P) = ∑_{Q∈P : Q≥P} β(Q),

β(P) = ∑_{r≥0}(−1)^{r}p^{r(r−1)/2} ∑_{Q∈P : Q/P ≅C_{p}^{r}} α(Q).

I cannot resist briefly digressing to show how Theorem 3.1 provides a very short proof of Virág’s recent generalisation [15] of Sylow’s Theorem.

Theorem 3.2: (Virág [15]) Let P ≤ G with |P | = p^{b}, and let a ≥ b with p^{a} dividing |G|. Then the number of subgroups Q of G containing P with |Q| = p^{a} is congruent to unity modulo p.

Proof: The case where b = 0 and a = 1 has long been known, indeed, it was essentially known to Cauchy, and anyway, is an immediate consequence of Frobenius’ theorem (again) in Curtis–Reiner [7, 41.11]. In general, for a p-subgroup U of G, define β(U) = 0 if |U| = p^{a}, and β(U) = 0 otherwise. Let α be as in Theorem 3.1(1). By induction on p^{a}/|U|, and Cauchy’s case above, α(U) ≡_{p} 1 when |U| ≤ p^{a}. ⊓ ⊔

Let Γ′(S, G) be the subset of Ω(G) consisting of the pairs (x, y) such that no non-trivial element of S is centralised by both x and y. Let

Δ′(S, G) := Ω(G) − Γ′(S, G) = ⋃_{P ∈S(S)}{(x, y) ∈ Ω(G):P = C_{S}(x) ∩ C_{S}(y)}.

Much as in Section 2, we define p-local integers γ′(G) := |Γ′(S, G)|/|G| and δ′(G) := |Δ′(S, G)|/|G|. Of course,

ω(G) = γ′(G) + δ′(G).

Given a finite elementary abelian p-group P, let r(P) denote the rank of P. Given a subgroup H ≤ G, let out_{G}(H) denote the index of the canonical image of N_{G}(H) in the automorphism group Aut(H). Thus

out_{G}(H) |N_{G}(H)| = |Aut(H)| |C_{G}(H)|.

Proposition 3.3: As p-local integers,

z_{+}(G) + γ′(p, G) ≡_{p} ∑_{Q}(−1)^{r(Q)+1}k(C_{G}(Q))/|N_{G}(Q):C_{G}(Q)|_{p′}

where Q runs over representatives of the G-conjugacy classes of non-trivial elementary abelian p-subgroups of G such that out_{G}(Q) is coprime to p.

Proof: For any subgroup P ≤ S put

α(P) := ω(C_{G}(P))|C_{G}(P)| = |{(x, y) ∈ Ω(G):P ≤ C_{S}(x) ∩ C_{S}(y)}|,
β(P) := |{(x, y) ∈ Ω(G):P = C_{S}(x) ∩ C_{S}(y)}|.

By Theorem 3.1(2),

δ′(G)|G| = α(1) − β(1) = ∑_{Q∈A(S)}(−1)^{r(Q)+1}p^{r(Q)(r(Q)−1)/2}α(Q)

= ∑_{Q∈_{G}A(G)}(−1)^{r(Q)+1}out_{G}(Q)_{p}|N_{G}(Q)|_{p}|C_{G}(Q)|_{p′} m_{G}(Q)ω(C_{G}(Q))

Theorem 1.1 and Remark 2.3 give

δ′(G) ≡_{p} ∑_{Q}(−1)^{r(Q)}k(C_{G}(Q))/|N_{G}(Q):C_{G}(Q)|_{p′}

where Q runs as in the assertion. Applying Theorem 1.1 again completes the argument. ⊓ ⊔

A sufficient condition for all the involutions in a finite 2-group P to be central is that P is the direct product of abelian and generalised quaternion groups. Again, further examples may be constructed using direct products.

Theorem 3.4: Suppose that p = 2, and that a Sylow 2-subgroup S of G has a unique maximal elementary abelian subgroup. Then z_{+}(G) has the same parity as the number of conjugacy classes of elementary abelian 2-subgroups Q of G such that k(C_{G}(Q)) and out_{G}(Q) are both odd. In particular, if all the involutions in S are central, then z_{+}(G) has the same parity as the number of conjugacy classes of involutions g in G such that k(C_{G}(g)) is odd.

Proof: We are to show that

z_{+}(G) ≡_{2} ∑_{Q} k(C_{G}(Q))

where Q runs as in Proposition 3.3. Let Γ := Γ′(S, G). We must show that 2|S| divides |Γ|. The demonstration now proceeds exactly as in the proof of Theorem 2.5. ⊓ ⊔

Examples for which γ′(G) is coprime to p abound when p ≥ 5. Thus the arguments in this section seem less susceptible to improvement than those in Section 2. However, I have been unable to find any such examples when p = 2 or p = 3, and in view of Proposition 4.1 below, propose that Theorem 3.4 is a “combinatorial shadow” of some general equality of sums of squares.

4: Formulas for the number of defect-zero irreducible characters

This appendant section, based on material in [3], Broué [5], Broué–Robinson [6], Robinson [15] bears on the material above only in the suggestive sense discussed in the previous paragraph. Following Broué [5], we define a function ψ_{p} on G × G such that ψ_{p}(g, h) is the number of elements z ∈ G for which gzh^{−1}z^{−1} is a p-element. Let ψ(g, h) be the number of solutions in x, y, z ∈ G to the equation gzh^{−1}z^{−1} = [x, y]. We define Ψ_{p}(G) := (ψ_{p}(g, h))_{g,h} and Ψ(G) := (ψ(g, h))_{g,h} as matrices indexed by representatives of the conjugacy classes of G. Parts of the following result — as indicated in the proof — are due to Broué and Robinson.

Proposition 4.1: (a) z_{0}(G) is the p-rank of Ψ_{p}(G).

(b) z_{0}(G) is the p-rank of Ψ(G).

(c) ψ_{p}(g, h) ≡_{p} ∑_{χ} χ(g)χ(h^{−1}) where g, h ∈ G, and χ runs over the defect-zero irreducible characters of G.

(d) ψ(g, h) = ∑_{χ}(|G|/χ(1))^{2}χ(g)χ(h^{−1}) where χ runs over all the irreducible characters of G.

(e) If p = 2 or p = 3, then Ψ_{p}(G) ≡_{p} Ψ(G).

Proof: Parts (a), (b), (d) are special cases of Broué–Robinson [6, 1.15], [3, 6(a)], [3, 3(b)], respectively. Part (e) will be immediate from (c) and (d). (We also note that part (a) may quickly be recovered from (c) by considering the basis of ZKG consisting of all the primitive idempotents e_{χ}.) Finally, to prove part (c), we persue a line of reasoning from Broué [5, page 385]. Let χ_{p} be the permutation character of G afforded by the cosets of the Sylow p-subgroup S. Then

|G:S|ψ_{p}(g, h) ≡_{p} χ_{p}(∑_{z∈G} gzh^{−1}z^{−1})

because χ_{p}/|G : S| is congruent to the characteristic function on the subset of G consisting of the p-elements. By considering the central character associated with any irreducible character χ of G, we obtain

χ_{p}(∑_{z∈G} gzh^{−1}z^{−1}) = |G|χ(g)χ(h^{−1})/χ(1).

If χ has defect-zero, then it vanishes on the non-trivial p-elements and therefore has multiplicity χ(1)/|S| in χ_{p}. The assertion follows. ⊓ ⊔

When the congruence in (e) holds, the formulas (a) and (b) for z_{0}(G) of course coincide; the congruence rarely holds when p ≥ 5.

Acknowledgements

This work benefited considerably from discussions with Burkhard Külshammer, Geoffrey R. Robinson, Jacques Thévenaz, and Jim Wiegold; I wish to thank them.

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Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey. e-mail: barker@fen.bilkent.edu.tr April 1997