THE NUMBER OF BLOCKS WITH A GIVEN DEFECT GROUP

Laurence Barker

August 1996

Abstract: Given a p-subgroup P of a finite group G, we express the number of p-blocks of G with defect group P as the p-rank of a symmetric integer matrix indexed by the N(P)/P -conjugacy classes in P C(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.

Question A: Given a p-subgroup P of a finite group G, how many p-blocks of G have defect group P? In particular, when is P a defect group in G?

An answer to this venerable question was given in Robinson [5], reformulated in Broué [2], and generalised and further illuminated in Broué–Robinson [3]. The number of p-blocks with defect group P is presented, in those three works, as the p-rank of a symmetric integer matrix indexed by certain conjugacy classes of G.

Question B: Supposing that G is a normal subgroup of a finite group F, what is the number f_{0}(G, F) of F/G-orbits of defect-zero p-blocks of G whose stabilisers in F/G are p′-groups?

Question B is more general than Question A because Brauer’s extended first main theorm describes a bijective correspondence between the p-blocks of G with defect group P, and the N_{G}(P)/P C_{G}(P)-orbits of defect-zero pblocks b of P C_{G}(P)/P such that the stabiliser of b in N_{G}(P)/P C_{G}(P) is a p′-group. In answer to Question B, Theorem 5 below expresses f_{0}(G, F) as the p-rank of a symmetric integer matrix Ψ(G, F) indexed by the F - conjugacy classes of G. Corollary 6 spells out the new answer thus provided to Question A.

In view of the local reduction indicated above, and also in view of another local reduction conjectured by Alperin [1], the number f_{0}(G) := f_{0}(G, G) of defect-zero p-blocks of G is of especial interest. A description of f_{0}(G) as the p-rank of a symmetric integer matrix indexed by the conjugacy classes of G has already been given by Robinson [5], but it is to be noted that the matrix Ψ(G) := Ψ(G, G) has the feature of being independent of p.

This work is a synthesis of character-theoretic constructions in Strunkov [6] and G-algebra-theoretic techniques implicit in Broué [2], Broué–Robinson [3], Robinson [5]. We also make use of a G-algebra-theoretic approach to Clifford theory. I am grateful to Robinson for communicating to me an illuminating formulation of material in [6].

Let O be a complete local noetherian commutative ring whose residue field O/J(O) has prime characteristic p, and whose field of fractions κ has characteristic zero. We shall assume that κ splits for all the given groups under consideration. When speaking of an integer, we shall always be refering to a rational integer, and we shall identify the integers with the elements of the minimal unital subring of O. Also, we shall identify the p-blocks of G with the block idempotents of OG. Let Irr(κG) denote the set of irreducible κG-characters, and Irr_{0}(κG) the subset consisting of those χ ∈ Irr(κG) such that |G|/χ(1) is coprime to p.

The conjugation action of F on its normal subgroup G induces algebra automorphisms of κG and OG, and induces permutations of Irr(κG). Recall that, for H ≤ K ≤ F, the image (OG)_{H}^{K} of the relative trace map Tr_{H}^{K} : (OG)^{H} → (OG)^{K} is an ideal of the K-fixed subalgebra (OG)^{K} of OG. For each irreducible κG-character χ, we write N_{F} (χ) for the stabiliser of χ in F, and write b_{χ}^{F} for the primitive idempotent of the commutative algebra (OG)^{F} such that χ(b_{χ}^{F}) = χ(1). Note that b_{χ}^{F} is the sum of the F -conjugates of the p-block b_{χ}^{G} of G containing χ. We say that a primitive idempotent b of (OG)^{F} is projective provided b ∈ (OG)_{1}^{F} (when G = F, these idempotents are precisely the defect-zero p-blocks of G).

Proposition 1: There is a bijective correspondence between the projective primitive idempotents b of (OG)^{F}, and the F -orbits of irreducible κGcharacters χ such that |N_{F} (χ)|/χ(1) is coprime to p, whereby b corresponds to the F -orbit of χ provided b = b_{χ}^{F}.

PROOF: Let χ be an irreducible κG-character, and put N := N_{F} (χ). It is well-known that b_{χ}^{G} is a defect-zero p-block of G if and only if |G|/χ(1) is coprime to p. When these equivalent conditions hold, the F -conjugates of χ are precisely the irreducible κG-characters χ′ such that b_{χ}^{F} = b_{χ′}^{F}. So it suffices to prove that b_{χ}^{F} is projective if and only if |N|/χ(1) is coprime to p.

Suppose that |N|/χ(1) is coprime to p. Then b_{χ}^{G} is a defect-zero block of OG, that is, b_{χ}^{G} = Tr_{1}^{G}(η) for some η ∈ OG. Also, |G: N| is coprime to p, and N is the stabiliser of b_{χ}^{F} in F, hence the primitive idempotent

b_{χ}^{F} = Tr_{N}^{F}(b_{χ}^{G}) = Tr_{1}^{F} (η/|G:N|)

is projective. Conversely, suppose that b_{χ}^{F} is projective. By Mackey decomposition,

(OG)_{1}^{F} ⊆ (OG)_{1}^{N} ⊆ (OG)_{1}^{G}.

So the idempotent b_{χ}^{G} = b_{χ}^{F} b_{χ}^{G} belongs to the ideal (OG)_{1}^{N} of (OG)^{N}, and in particular, b_{χ}^{G} is a defect-zero block of OG. Therefore, |G|/χ(1) is coprime to p, and writing b_{χ}^{G} = Tr_{1}^{N}(μ) for some μ ∈ OG, we have

b_{χ}^{G} = Tr_{1}^{N}(μb_{χ}^{G}) = |N:G|Tr_{1}^{G}(μb_{χ}^{G})

whereupon |N:G| must be coprime to p. ⊓ ⊔

In particular, f_{0}(G, F) is the number of projective primitive idempotents of (OG)^{F}. Let b_{G}^{F} be the sum of the projective primitive idempotents of (OG)^{F}. Thus b_{G}^{G} is the sum of the defect-zero p-blocks of G. We have a direct sum of free cyclic O-modules

(OG)^{F} b_{G}^{F} = ⨁_{b}(OG)^{F} b

where b runs over the projective primitive idempotents of (OG)^{F}. IIzuka– Watanabe [4, Lemma 2] proved the following result in the special case G = F.

Lemma 2: (OG)^{F} b_{G}^{F} ⊆ ((OG)_{1}^{F})^{2} ⊆ (OG)^{F} b_{G}^{F} ⊕ J(O)(OG)^{F} (1 − b_{G}^{F}).

PROOF: Let V := (OG)^{F} b_{G}^{F} and U := ((OG)_{1}^{F})^{2} as ideals of (OG)^{F}. Clearly, b_{G}^{F} ∈ V, so V ⊆ U. Now

U = ⨁_{χ∈_{F}Irr(κG)} Ub_{χ}^{F}

where the notation indicates that χ runs over the F -orbits of Irr(κG). Let us fix an irreducible κG-character χ. If b_{χ}^{F} is projective, then b_{χ}^{F} ∈ V, so Ub_{χ}^{F} ⊆ V. Assuming now that b_{χ}^{F} is not projective, we have b_{G}^{F}b_{χ}^{F} = 0, and it suffices to show that Ub_{χ}^{F} ⊆ J(O)(OG)^{F}.

First suppose that χ ∈ Irr_{0}(κG). Then b_{χ}^{F} ∉ U, so Ub_{χ}^{F} is strictly contained in the free cyclic O-module (OG)^{F} b_{χ}^{F}. Therefore Ub_{χ}^{F} ⊆ J(O)(OG)^{F} in this case. Now suppose that χ ∉ Irr_{0}(κG). We observed above that (OG)_{1}^{F} ⊆ (OG)_{1}^{G}, and that the assertion holds when G = F. So

Ub_{χ}^{F} ⊆ U ⊆ ((OG)_{1}^{G})^{2} ⊆ ZOG.b_{G}^{G} ⊕ J(O).ZOG(1 − b_{G}^{G}).

But b_{χ}^{G}b_{χ}^{F} = 0, so Ub_{χ}^{F} ⊆ J(O).ZOG ∩ (OG)^{F} = J(O)(OG)^{F}. ⊓ ⊔

For each irreducible κG-character χ, let e_{χ}^{F} be the primitive idempotent of (κG)^{F} such that χ(e_{χ}^{F}) = χ(1), and let ω_{χ}^{F} be the algebra map (κG)^{F} → κ such that ω(e_{χ}^{F}) = 1. Then ω_{χ}^{F} is a restriction of the central character ω_{χ}^{G} associated with χ. That is to say, ω_{χ}^{F} = χ/χ(1) on (κG)^{F}. Let ω_{G}^{F} : (κG)^{F} → κ, and ν_{G}^{F} : (κG)^{F} ⊗_{κ}(κG)^{F} → κ be the characters afforded by the translation actions of (κG)^{F} and (κG)^{F} ⊗_{κ} (κG)^{F} on (κG)^{F}. Thus, given ζ, ζ′ ∈ (κG)^{F}, the trace of the action of ζ on (κG)^{F} is

ω_{G}^{F}(ζ) = ∑_{χ∈_{F}Irr(κG)} ω_{χ}^{F} (ζ)

while the trace of the action of ζ ⊗ ζ′ on (κG)^{F} is

ν_{G}^{F}(ζ ⊗ ζ′) = ω_{G}^{F}(ζζ′) = ∑_{χ∈Irr(κG)} ω_{χ}^{F} (ζ)ω_{χ}^{F} (ζ′).

We define linear maps ϕ_{G}^{F} : κG → κ and ψ_{G}^{F} : κ(G × G) → κ given by

ϕ_{G}^{F}(g) := |{(x, w) ∈ G × F : g = [x, w]}|,
ψ_{G}^{F}(g, h) := |{(x, w, v) ∈ G × F × F : gvh^{−1}v^{−1} = [x, w]}|

for g, h ∈ G. Here, [x, w] := xwx^{−1}w^{−1}. The next result shows that ϕ_{G}^{F} is a κG-character, and that ψ_{G}^{F} is a κ(G × G)-character. Recall that the irreducible κ(G × G)-characters are the κ(G × G)-characters of the form χ ∗ χ′ with χ, χ′ ∈ Irr(κG), where

(χ ∗ χ′)(g, h) := χ(g)χ′(h^{−1}).

For any element x ∈ G, we write [x]_{F} for the set of F -conjugates of x. For any subset A ⊆ G, we write A^{+} for the sum in κG of the elements of [x]_{F}. Thus {[x]_{F}^{+} : x ∈ G} is a κ-basis for (κG)^{F}.

Lemma 3: (a) ϕ_{G}^{F} = ∑_{χ∈Irr(κG)} |N_{F} (χ)| χ/χ(1).

(b) ψ_{G}^{F} = ∑_{χ∈Irr(κG),vN_{F} (χ)⊆F} (|N_{F} (χ)|/χ(1))^{2} χ ∗^{v}χ.

PROOF: Given an element ζ ∈ (κG)^{F}, and writing

ζ[x]_{F}^{+} = ∑_{y∈_{F} G} ζ_{x,y}[y]_{F}^{+}

then ω_{G}^{F}(ζ) = ∑_{y∈_{F} G} ζ_{y,y}, where the notation indicates that the two sums are indexed by representatives y of the F -conjugacy classes of G. So for g, h ∈ G, we have

ψ_{G}^{F}(g, h) = |{(x, w, v, u) ∈ G × F × F × F : ugu^{−1}vh^{−1}v^{−1}wxw^{−1} = x}|/|F |

= ∑_{x∈_{F} G} |{(w, v, u) ∈ F × F × F : ugu^{−1}vh^{−1}v^{−1}wxw^{−1} = x}|/|C_{F} (x)|

= ∑_{x∈_{F} G} |{(y, v, u) ∈ [x]_{F} × F × F : ugu^{−1}vh^{−1}v^{−1}y = x}|

= ν_{G}^{F}(Tr_{1}^{F} (g) ⊗ Tr_{1}^{F} (h^{−1}))

= ∑_{χ∈_{F}Irr(κG)} χ(Tr_{1}^{F} (g))χ(Tr_{1}^{F} (h^{−1}))/χ(1)^{2}

= |G|^{2} ∑_{χ∈_{F}Irr(κG),uG⊆F,vG⊆F} χ(ugu^{−1})χ(vh^{−1}v^{−1})/χ(1)^{2}

= ∑_{χ∈_{F}Irr(κG),vN_{F} (χ)⊆F}(|N_{F} (χ)|/χ(1))^{2}χ(ugu^{−1})χ(vh^{−1}v^{−1}).

Part (b) is thus established. Part (a) may be proved either by showing, similarly, that

ϕ_{G}^{F}(g) = ω_{G}^{F}(Tr_{1}^{F} (g)) = ∑_{χ∈Irr(κG)} |N_{F} (χ)| χ(g)/χ(1)

or else by observing that ϕ_{G}^{F}(g) = ψ_{G}^{F}(g, 1)/|F |. ⊓ ⊔

Lemma 4: Given g, h ∈ G, then:

(a) |C_{F} (g)| divides ϕ_{G}^{F}(g),

(b) |C_{F} (g)||C_{F} (h)| divides ψ_{G}^{F}(g, h).

PROOF: The proof of Lemma 3 shows that ϕ_{G}^{F}(g)/|C_{F} (g)| = ω_{G}^{F}([g]_{F}^{+}), and ψ_{G}^{F}(g, h)/|C_{F} (g)||C_{F} (h)| = ν_{G}^{F}([g]_{F}^{+}⊗[h^{−1}]_{F}^{+}). Any rational number belonging to O must be an integer. ⊓ ⊔

We define a symmetric integer matrix Ψ(G, F) := (ψ_{G}^{F}(g, h))_{g,h∈_{F} G} (indexed by representatives of the F -conjugacy classes in G). For any integer matrix Ψ, let rk_{p}Ψ denote the p-rank of Ψ (the rank of the reduction of Ψ modulo p).

Theorem 5: (a) f_{0}(G, F) = rk_{p}Ψ(G, F).

(b) (OG)^{F} has a projective primitive idempotent if and only if ϕ_{G}^{F}(x) is coprime to p for some x ∈ G.

PROOF: Any projective primitive idempotent b of (OG)^{F} may be written in the form b = Tr_{1}^{F} (∑_{g∈G} b_{g}g) with each b_{g} ∈ O. Given another projective primitive idempotent b′ of (OG)^{F}, the proof of Lemma 3 gives

ψ_{G}^{F}(∑_{g∈G} b_{g}g ⊗ ∑_{h∈G} b′_{h}h^{−1}) = ν_{G}^{F}(b ⊗ b′) = ω_{G}^{F}(bb′)

which is zero when b ≠ b′. But ω_{G}^{F}(b) = 1 because b = e_{χ}^{F} for some χ ∈ Irr_{0}(κG). So f_{0}(G, F) ≤ rk_{p}Ψ_{G}^{F}. The proof of Lemma 3 also shows that

ψ_{G}^{F}(g, h) = ω_{G}^{F}(Tr_{1}^{F} (g)Tr_{1}^{F} (h^{−1}))

for g, h ∈ G. But f_{0}(G, F) is the O-rank of (OG)^{F} b_{G}^{F}, so Lemma 2 forces f_{0}(G, F) ≥ rk_{p}Ψ_{G}^{F}, establishing part (a).

Part (b) is the assertion that ψ_{G}^{F}(g, h) is coprime to p for some g, h ∈ G if and only if ϕ_{G}^{F}(x) is coprime to p for some x ∈ G. The forwards implication holds because ψ_{G}^{F}(g, h) = ϕ_{G}^{F}(gTr_{1}^{F} (h^{−1})) for all g, h ∈ G. Proposition 1 and Lemma 3(a) give the reverse implication. ⊓ ⊔

Thanks to Lemma 4(b), f_{0}(G, F) is, in fact, the p-rank of the submatrix of Ψ(G, F) indexed by those representatives g such that C_{F} (g) is a p′-group.

Corollary 6: Let P be a p-subgroup of G. Define \overline{C} := P C_{G}(P)/P and \overline{N} := N_{G}(P)/P. Then:

(a) The number of p-blocks of G with defect group P is rk_{p}Ψ(\overline{C}, \overline{N}).

(b) P is a defect group of a p-block of G if and only if there exists some g ∈ \overline{C} such that p does not divide the number of solutions in x ∈ \overline{C} and w ∈ \overline{N} to the equation g = [x, w].

PROOF: This is immediate from Proposition 1, Theorem 5, and Brauer’s extended first main theorem. ⊓ ⊔

Putting G = F in Theorem 5, or P = 1 in Corollary 6, we deduce that f_{0}(G) is the p-rank of the symmetric integer matrix Ψ(G) := Ψ(G, G) indexed by the representatives of the conjugacy classes of G. We note, as above, that to calculate f_{0}(G), we need only consider the submatrix indexed by the representatives of the defect-zero conjugacy classes. Furthermore, we recover the special case of Strunkov [6, Theorem 1] asserting that G has a defect-zero p-block if and only if there exists an element g ∈ G such that p does not divide the number of ways of expressing g as a commutator [x, w] with x, w ∈ G.

We end by observing another relationship between the character ϕ_{G} := ϕ_{G}^{G} and the number of defect-zero p-blocks of G. Let us fix a Sylow p-subgroup S of G. We write (Res_{S}^{G}(ϕ_{G}), 1_{S}) to denote the multiplicity of the trivial κS-module in the restriction of ϕ_{G} to S. Note that

(Res_{S}^{G}(ϕ_{G}), 1_{S}) = |{(x, w) ∈ G × G : [x, w] ∈ S}|/|S|.

Proposition 7: Modulo p, we have a congruence

|G : S|f_{0}(G) ≡_{p} (Res_{S}^{G}(ϕ_{G}), 1_{S}).

PROOF: For any ζ ∈ ZOG, Lemma 3(a) gives

ϕ_{G}(ζ) = |G| ∑_{χ∈Irr(κG)} ω_{χ}(ζ).

In particular, |G| divides ψ_{G}(ζ). Let G_{p} denote the set p-elements of G. By IIzuka–Watanabe [4, Lemmas 3 and 4], if χ ∈ Irr_{0}(κG) then ω_{χ}(G_{p}^{+}) = 1, otherwise p divides ω_{χ}(G_{p}^{+}). So

f_{0}(G) ≡_{p} ϕ_{G}(G_{p}^{+})/|G|.

Let N := N_{G}(S). A well-known variant of Sylow’s theorem asserts that the number of Sylow p-subgroups of G containing any given p-subgroup is congruent to unity modulo p. Applying this to the cyclic p-subgroups, we deduce that |G : N| is congruent to unity modulo p, and so too is the coefficient in Tr_{N}^{G}(S^{+}) of each p-element. Hence (Tr_{N}^{G}(S^{+}) − G_{p}^{+})/p ∈ ZOG, and p|G| divides ϕ_{G}(Tr_{N}^{G}(S^{+}) − G_{p}^{+}). Therefore

ϕ_{G}(G_{p}^{+})/|S| ≡_{p} ϕ_{G}(Tr_{N}^{G}(S^{+}))/|S| ≡_{p} ϕ_{G}(S^{+})/|S| = (Res_{S}^{G}(ϕ_{G}), 1_{S}). ⊓ ⊔

References

[1] J. L. Alperin. Weights for finite groups. Proc. Symp. Pure Math. 47 (1987), 369-379.

[2] M. Broué. On a theorem of G. Robinson. J. London Math. Soc. 29 (1984), 425-434.

[3] M. Broué and G. R. Robinson. Bilinear forms on G-algebras. J. Algebra 104 (1986), 377-396.

[4] K. Iizuka and A. Watanabe. On the number of blocks of irreducible characters of a finite group with a given defect group. Kumamoto J. Sci. (Math.) 9 (1973), 55-61.

[5] G. R. Robinson. The number of blocks with a given defect group. J. Algebra 84 (1983), 493-502.

[6] S. P. Strunkov. Existence and the number of p-blocks of defect 0 in finite groups. Algebra and Logic 30 (1992), 231-241.

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey.

20C20: REPRESENTATION THEORY OF GROUPS; Modular representations and characters.