SYLOW INTERSECTIONS, DOUBLE COSETS, AND 2-BLOCKS

J. Murray* Mathematics Department, University College Dublin, Belfield Dublin 4, Ireland.

1. Notation and statement of results

Throughout G will be a finite group and F will be a finite field of characteristic p > 0, although we are mainly interested in the case p = 2. For convenience we assume that F is a splitting field for all subgroups of G. We let ℤ_{(p)} denote the localization of the integers ℤ at the prime ideal pℤ. If x ∈ ℤ_{(p)}, then x^{∗} will denote its image modulo the unique maximal ideal of ℤ_{(p)}. We regard x^{∗} as lying in the prime field GF(p) of F.

The elements of G may be identified with the members of a distinguished basis of the group algebra F G. Thus each x ∈ F G is of the form x = ∑_{g∈G} β(x, g)g, where β(x, g) ∈ F, for g ∈ G. We define the element x^{o} of F G as x^{o} := ∑_{g∈G} β(x, g^{−1})g. The map x → x^{o} is an anti-isomorphism of F G, and its restriction to the centre Z of F G is an involutary isomorphism. We use 𝒦^{+} to denote the sum of the elements in a G-conjugacy class 𝒦 in F G.

1991 Mathematics Subject Classification. 20C20.

*The author was supported by an Enterprise Ireland research grant while writing this paper.

The set of all such class sums forms an F -basis for Z. If x ∈ Z and g ∈ 𝒦, we will use β(x, 𝒦^{+}) in place of β(x, g).

By a p-block B of G we mean a direct F -algebra summands of F G. Associated with B there is a primitive idempotent e ∈ Z, and an F -epimorphism ω : Z → F. We indicate these associations by B ↔ e ↔ ω, and call e the block idempotent, and ω the central character, of B. Set B^{o} := {x ∈ F G | x^{o} ∈ B} and ω^{o}(x) := ω(x^{o}). Then B^{o} ↔ e^{o} ↔ ω^{o} is a p-block of G. We say that B is a real block of G if B = B^{o}.

R. Brauer showed how to associate with B a G-conjugation family of pgroups, which he called the defect groups of B. Let D be a defect group of B. Then D is not arbitrarily embedded in G. For instance Brauer proved that D is the largest normal p-subgroup of its normalizer N(D). J. A. Green [5] showed that there exists g ∈ G and a Sylow p-subgroup S of G, such that S ∩ S^{g} = D, and M. F. O’Reilly [9] showed that g could be chosen to be p-regular with defect group D. Here a defect group of g means a Sylow p-subgroup of the centralizer C(g) of g in G.

Let e_{D} denote the sum of the block idempotents associated with the p-blocks of G which have defect group D, and let 𝒮yl denote the collection of Sylow p-subgroups of G. We prove the following partial converse to these results:

Theorem 2.9. Let p = 2 and let g ∈ G be 2-regular with defect group D. Then β(e_{D}, g) = |{P ∈ 𝒮yl | P ∩ P^{g} = D, P gP = P g^{−1}P }| 1_{F}.

Now G has a 2-block with defect group D if and only if β(e_{D}, g) ≠ 0 for some 2-regular element g with defect group D. So 2.9 furnishes a necessary and sufficient condition for G to have a 2-block with defect group D.

If g ∈ G, set C^{∗}(g) := {x ∈ G | g^{x} ∈ {g, g^{−1}}}. We call the Sylow 2subgroups of C^{∗}(g) the extended defect groups of g. Let 𝒦 be the conjugacy class of G. The extended defect groups of the elements of 𝒦 form a single G-orbit. which we call the extended defect groups of 𝒦. If E is an extended defect group of g, then D := C_{E}(g) is a group of g which is contained in E, and |E : D| ≤ 2. We call (D, E) a defect pair for g. The defect pairs of the elements of 𝒦 form a single G-orbit. We call g a real element if it is G-conjugate to g^{−1}. Theorem 2.9 can be refined for real elements as follows:

Theorem 3.1. Let p = 2 and let g ∈ G be real and 2-regular with defect pair (D, E). Then β(e_{D}, g) = |{P ∈ 𝒮yl | P ∩ P^{g} = D, E ≤ P }| 1_{F}.

We use this theorem to give an alternative proof of Theorem 4.8 of [4] and also to provide a self-contained treatment of some results of M. Herzog.

Let 𝒦 be a conjugacy class of G. We call 𝒦 a real class if it coincides with its inverse class 𝒦^{o} := {g ∈ G | g^{−1} ∈ 𝒦}. We call a real class 𝒦 properly real if g^{2} ≠ 1 for g ∈ 𝒦.

Suppose that 𝒦 and ℒ are conjugacy classes of G. We write

𝒦 ≤ ℒ,

if each defect group of 𝒦 is contained in some defect group of ℒ. Suppose in addition that 𝒦 is properly real and that ℒ is real. We write

𝒦 ⪯ ℒ, (1.1)

if for each defect pair (D, E) of 𝒦, there exists l ∈ ℒ such that D ≤ C(l) and E ≤ C^{∗}(l), but E ≰ C(l) if l^{2} ≠ 1.

Let B ↔ e ↔ ω be a 2-block of G. It is well-known that there exists a 2-regular class ℒ of G such that β(e, ℒ^{+}) ≠ 0 and ω(ℒ^{+}) ≠ 0. Any such ℒ is called a defect class for B. The min-max theorem [7, 15.31] states that

\begin{aligned}ω(𝒦^{+}) ≠ 0 & ⟹ ℒ ≤ 𝒦, \\ β(e, 𝒦^{+}) ≠ 0 & ⟹ 𝒦 ≤ ℒ, \end{aligned} for each class 𝒦 of G. (1.2)

Suppose that B is a real 2-block of G. R. Gow showed in [3] that B has a defect class 𝒦 which is real. Let (D, E) be a defect pair for 𝒦. Gow proved that the extended defect groups of the real defect classes of B are G-conjugate to E. For this reason he referred to the G-conjugates of E as the extended defect groups of B. We call (D, E) a defect pair for B. Theorem 2.1 of [3] can be extended in the following way:

Theorem 4.3 (Min-Max for Real 2-Blocks). Let B ↔ e ↔ ω be a real nonprincipal 2-block of G and let ℒ be a real defect class of B. Then

\begin{aligned}ω(𝒦^{+}) ≠ 0 & ⟹ ℒ ⪯ 𝒦, \\ β(e, 𝒦^{+}) ≠ 0 & ⟹ 𝒦 ⪯ ℒ, \end{aligned} for each real class 𝒦 of G.

Throughout the paper S will be a fixed Sylow 2-subgroup of G. Let D ≤ E be subgroups of S with |E : D| = 2. Let S\G/S denote a set of representatives for the double cosets of S in G. If x, y ∈ G lie in the same (S, S)-double coset, then the groups S ∩ S^{x} and S ∩ S^{y} are S-conjugate. We say that SgS is a self-dual double coset if SgS = Sg^{−1}S. Lemma 5.1 furnishes a 2-subgroup (S ∩ S^{g})^{∗} of G which contains S ∩ S^{g} as a subgroup of index 2, whenever SgS is self-dual and distinct from S. Moreover, if x, y ∈ SgS, then (S ∩ S^{x})^{∗} and (S ∩ S^{y})^{∗} are conjugate in S.

If H and K are subgroups of G, we write H =_{G} K if some G-conjugate of H equals K, and we write H ≤_{G} K if some G-conjugate of H is contained in K. We let ∑_{x}^{D} denote a sum which ranges over those elements x of S\G/S for which S ∩ S^{x} =_{G} D, and let ∑_{x≡x^{−1}}^{D} denote the restriction of this sum to the self-dual double cosets.

Suppose that {𝒦_{1}, …, 𝒦_{v}} is a complete list of the real 2-regular classes of G which have defect pair (D, E). We call a self-dual double coset SgS a (D, E)-double coset if there exists x ∈ Sg which simultaneously satisfies:

(1) x ∈ 𝒦_{1} ⋃ ⋯ ⋃ 𝒦_{v};

(2) (S ∩ S^{g}, (S ∩ S^{g})^{∗}) is a defect pair for x. Let x_{1}, …, x_{w} be a (possibly empty) set of representatives for the (D, E)- double cosets of S. Suppose that w ≠ 0. We define an v × w integer matrix N by setting the i, j-th entry of N to be the number N_{ij} of y_{i} in 𝒦_{i} ∩ Sx_{j} such that (S ∩ S^{x_{j}}, (S ∩ S^{x_{j}})^{∗}) is a defect pair for y_{i}. When p = 2, Theorem A of [11] can be refined as follows:

Theorem 5.2. The number of real 2-blocks of G which have defect pair (D, E) is zero, if w = 0, and is the 2-rank of the matrix N · N^{T}, if w ≠ 0.

2. Sylow intersections and 2-blocks

Our starting point is Proposition 3.1 of [4]:

Lemma 2.1. Let B ↔ e ↔ ω be a p-block of G which has defect group D and let 𝒦 be a p-regular class of G which has defect group D. Then

β(e, 𝒦^{+}) = (\frac{dim(B)}{|G| |𝒦|})^{∗} ω(𝒦^{o+}).

Let JZ denote the Jacobson radical of Z and let EZ denote the F -span of the idempotents in Z. Then JZ is an ideal of Z, EZ is a direct sum of copies of F (as an F -algebra) and Z = JZ ⨁ EZ as F -algebras. The Robinson map is the natural F -algebra projection ϵ : Z → EZ with respect to this decomposition. Let z ∈ Z and let n be a positive integer such that g^{p^{n}} = 1_{G}, for each p-element g of G, and λ^{p^{n}} = λ, for each λ ∈ F. Then ϵ(z) = z^{p^{n}}. We also have

ϵ(z) = ∑_{B} ω(z)e, (2.2)

where B ↔ e ↔ ω ranges over the p-blocks of G. See [12] for further details.

For the rest of the paper we take p = 2 and Char(F) = 2. Recall that e_{D} denotes the sum of the block idempotents in Z which have defect group D. We combine Lemma 2.1 and (2.2) as follows:

Corollary 2.3. Let 𝒦 be a 2-regular class of G which has defect group D. Then β(e_{D}, 𝒦^{+}) = β(ϵ(𝒦^{o+}), 𝒦^{+}). Proof. It follows from (1.2) and (2.2) that

β(ϵ(𝒦^{o+}), 𝒦^{+}) = ∑_{B} ω(𝒦^{o+})β(e, 𝒦^{+}),

where B ↔ e ↔ ω ranges over the 2-blocks of G which have defect group D. Also ( dim(B)/|G||𝒦|)^{∗} = 1_{F}, for each such B. Thus

β(ϵ(𝒦^{o+}), 𝒦^{+}) = ∑ β(e, 𝒦^{+})^{2}, using Lemma 2.1
= (∑ β(e, 𝒦^{+}))^{2}, as F has characteristic 2
= β(e_{D}, 𝒦^{+})^{2}
= β(e_{D}, 𝒦^{+}),  as β(e_{D}, 𝒦^{+}) ∈ GF(2).

If 𝒦 and ℒ are 2-regular classes which have defect group D then

β(ϵ(𝒦^{+}), ℒ^{+}) = ∑_{x}^{D} |𝒦 ∩ Sx| |ℒ ∩ Sx| 1_{F} , (2.4)

using 1.3.3 and 1.3.4 of [12]. This allows us to prove: Proposition 2.5. Let 𝒦 be a 2-regular class with defect group D. Then

β(e_{D}, 𝒦^{+}) = ∑_{x≡x^{−1}}^{D} |𝒦 ∩ Sx| 1_{F}.

Proof. By Corollary 2.3 and (2.4) we have

β(e_{D}, 𝒦^{+}) = ∑_{x}^{D} |𝒦^{o} ∩ Sx| |𝒦 ∩ Sx| 1_{F} . (2.6)

Let x ∈ G. The map sx ↔ (sx)^{−s}, for s ∈ S, establishes a bijection between the sets 𝒦 ∩ Sx and 𝒦^{o} ∩ Sx^{−1}. So |𝒦 ∩ Sx| = |𝒦^{o} ∩ Sx^{−1}|. (2.7)

Suppose SxS ≠ Sx^{−1}S. Then the contribution of these cosets to (2.6) is |𝒦^{o}∩Sx| |𝒦∩Sx|1_{F} +|𝒦^{o}∩Sx^{−1}| |𝒦∩Sx^{−1}|1_{F} = 2·|𝒦^{o}∩Sx| |𝒦∩Sx|1_{F} = 0_{F} . It follows that

β(e_{D}, 𝒦^{+}) = ∑_{x≡x^{−1}}^{D} |𝒦^{o} ∩ Sx| |𝒦 ∩ Sx| 1_{F} . (2.8)

Suppose that SxS = Sx^{−1}S. Then

|𝒦^{o} ∩ Sx| = |𝒦^{o} ∩ Sx^{−1}|, as Sx^{−1} and Sx are S-conjugate
= |𝒦 ∩ Sx|,  by (2.7).

We conclude from (2.8) and the fact that the prime field of F is GF(2) that

β(e_{D}, 𝒦^{+}) = ∑_{x≡x^{−1}}^{D} |𝒦 ∩ Sx|^{2} 1_{F} = ∑_{x≡x^{−1}}^{D} |𝒦 ∩ Sx| 1_{F} .

Proof of Theorem 2.9. Recall that g is a 2-regular element of G with defect group D. Let 𝒦 be the class of G which contains g. We shall compute |{(k, P) ∈ 𝒦 × 𝒮yl | P kP = P k^{−1}P, P ∩ P^{k} =_{G} D}| in two different ways. On the one hand it equals |𝒦| |μ(g)|, where

μ(g) := {P ∈ 𝒮yl | P gP = P g^{−1}P, P ∩ P^{g} =_{G} D}.

On the other hand it equals |𝒮yl| ∑_{x≡x^{−1}}^{D} |𝒦∩SxS|. The double coset SxS is a union of |S : S ∩S^{x}| right cosets of S, and each of these is S-conjugate to Sx. It follows that |𝒦 ∩ SxS| = |S : S ∩ S^{x}||𝒦 ∩ Sx|. Also |S : S ∩ S^{x}| = |S : D|, whenever S ∩ S^{x} is G-conjugate to D. But |𝒮yl| is odd, by Sylow’s Theorem. Thus

|μ(g)| 1_{F} = \frac{|S : D|}{|𝒦|} ∑_{x≡x^{−1}}^{D} |𝒦 ∩ Sx| 1_{F}
= ∑_{x≡x^{−1}}^{D} |𝒦 ∩ Sx| 1_{F} , as 𝒦 has defect group D
= β(e_{D}, g), by Proposition 2.5.

We claim that D acts by conjugation on μ(g). For, suppose that P ∈ μ(g) and d ∈ D. Then dg = gd. So P^{d} ∩ P^{dg} = (P ∩ P^{g})^{d} =_{G} D, and P^{d}gP^{d} = (P gP)^{d} = (P g^{−1}P)^{d} = P^{d}g^{−1}P^{d}. Thus P^{d} ∈ μ(g), which proves our claim.

Each D-orbit in μ(g) has 2-power order, and P is stabilized by D if and only if D ≤ P. But D ≤ P implies that D ≤ P ∩ P^{g}. Since P ∩ P^{g} =_{G} D, it follows that P is stabilized by D if and only if P ∩ P^{g} = D. We conclude that

|μ(g)| ≡ | {P ∈ 𝒮yl | P ∩ P^{g} = D, P gP = P g^{−1}P }| (mod2),

from which the theorem follows.

Theorem 2.9 has no obvious analogue for odd primes. For instance, if P is a Sylow 3-subgroup of PSL_{3}(2) and g is an element of order 4, the set {P ∈ 𝒮yl | P ∩ P^{g} = {1}, P gP = P g^{−1}P } has cardinality 4. However, g has 3-defect zero and appears with zero multiplicity in the sum of the 3-block idempotents of defect zero.

We indicate how our methods may be used to sharpen Corollary 2 of [11]:

Theorem 2.10. Let g be a 2-regular element of G which has defect group D. Suppose that P ∩ P^{g} = D, for each Sylow 2-subgroup P of G which contains D. Then g lies in a defect class of some real 2-block of G. In particular, G has a real 2-block with defect group D.

Proof. Let r_{D} denote the sum of the real 2-block idempotents of G which have defect group D, and let 𝒦 be the conjugacy class of G which contains g. We can show that

β(r_{D}, 𝒦^{+}) = β(ϵ(𝒦^{+}), 𝒦^{+}),

by modifying the proof of Corollary 2.3. We can then adapt the proofs of Proposition 2.5 and Theorem 2.9 to show that

β(r_{D}, g) = |{P ∈ 𝒮yl | P ∩ P^{g} = D}| 1_{F} . (2.11)

The number of Sylow 2-subgroups of G which contain D is odd, by a well known generalization of Sylow’s Theorem. It then follows from our hypothesis, and (2.11), that β(r_{D}, g) = 1_{F}. So G has a real 2-block B ↔ e ↔ ω which has defect group D, and β(e, 𝒦^{+}) = β(e, g) ≠ 0_{F}. Also ω(𝒦^{+}) = ω(𝒦^{o+}) ≠ 0_{F}, by Lemma 2.1. This completes the proof.

3. Real 2-regular classes and 2-blocks

In this section we prove Theorem 3.1 and give a number of applications.

Proof of Theorem 3.1. Recall that g is a 2-regular element of G with defect pair (D, E). Note that if E ≤ P, then P gP = P g^{−1}P.

We claim that E acts on the set ϕ(g) := {P ∈ 𝒮yl | P ∩ P^{g} = D, P gP = P g^{−1}P } by conjugation. For, suppose that P ∈ ϕ(g). Then D normalizes P. If e ∈ E\D then g^{e} = g^{−1}. So P^{e}gP^{e} = (P g^{−1}P)^{e} = (P gP)^{e} = P^{e}g^{−1}P^{e}. Moreover eg = g^{−1}e normalizes D. Thus P^{e}∩P^{eg} = (P^{g}∩P)^{g^{−1}e} = D^{g^{−1}e} = D. This shows that P^{e} ∈ ϕ(g), which proves our claim.

Each E-orbit on ϕ(g) has cardinality 1 or 2. Since P is a Sylow 2-subgroup of G, it is stablized by E if and only if E ≤ P. We conclude that

|ϕ(g)| ≡ |{P ∈ 𝒮yl | P ∩ P^{g} = D, E ≤ P }| (mod2).

The result now follows from Theorem 2.9.

In our first application of Theorem 3.1, we give another proof of [4, 4.8].

Theorem 3.2. Let D be a 2-subgroup of G. Suppose that no subgroup of N(D)/D is isomorphic to a dihedral group of order 8. Then β(e_{D}, g) = 1_{F}, for each real 2-regular element g of G which has defect group D. In particular, the following are equivalent: (a). G has a real 2-regular element with defect group D; (b). G has a 2-block with defect group D; (c). G has a real 2-block with defect group D.

Proof. The implications (c) ⟹ (b) ⟹ (a) follow as in [4, 4.8].

Suppose that D is a Sylow 2-subgroup of G. Then the principal 2-block B_{0} ↔ e_{0} ↔ ω_{0} is the only real block with defect group D, and the identity class is the only real 2-regular class with defect group D. Also β(e, 1^{+}) = β(e^{o}, 1^{+}), for each non-real 2-block idempotent. It follows that β(e_{D}, 1^{+}) = β(e_{0}, 1^{+}) = ω_{0}(1^{+}) = 1_{F}, using Lemma 2.1 (this also follows from a theorem of R. Brauer).

Suppose that D is not a Sylow 2-subgroup of G. Let g be a real 2-regular element with defect pair (D, E). The first statement and the implication (a) ⟹ (c) will follow from Theorem 3.1, if we can show that P ∩ P^{g} = D, whenever P is a Sylow 2-subgroup of G which contains E.

Assume for the sake of contradiction that there exists P ∈ 𝒮yl with E ≤ P and P ∩P^{g} > D. Let x ∈ E\D, and set y := x^{−1}g = g^{−1}x^{−1}. Then y ∈ N(D), since x ∈ E ≤ N(D) and g ∈ C(D). Also y ∈ N(P ∩ P^{g}), since (P ∩ P^{g})^{y} = P^{x^{−1}g} ∩ P^{x^{−1}} = P^{g} ∩ P.

The 2-group ⟨y⟩ acts on the nontrivial 2-group N_{P ∩P^{g}}(D)/D. So we can choose n ∈ N_{P ∩P^{g}}(D)\D such that n^{2} ∈ D and [n, y] ∈ D. Our hypothesis on N(D)/D forces [n, x] ∈ D. Therefore [n, g] = [n, xy] = [n, x][n, y]^{x^{−1}} lies in D. It follows that g centralizes ⟨D, n⟩, since ⟨g⟩ is a 2’-group which acts trivially on every factor of 1 ≤ D < ⟨D, n⟩. This contradicts the fact that D is a defect group of g. The theorem follows.

In our next application we give self contained proofs of a number of results on extremal 2-blocks which are due to M. Herzog [6]. We call G a CI-group if every intersection of distinct Sylow 2-subgroups of G is centralized by some Sylow 2-subgroup of G. It is straightforward to show that every subgroup and factor group of a CI-group is a CI-group. Let S and T be Sylow 2-subgroups of G. We say that S ∩ T is a maximal Sylow intersection in G if S ≠ T and whenever S ∩ T ≤ P ∩ Q, where P ≠ Q are Sylow 2-subgroups of G, then S ∩ T = P ∩ Q.

Lemma 3.3. Let G be a CI-group. Suppose that S ≠ T are Sylow 2-subgroups of G. Then S ∩ T is centralized by every 2-group which contains it.

Proof. Let R be a 2-subgroup of G which contains S ∩ T. Since S ≠ T, we may assume that R ≠ T. Then R ∩ T ≥ S ∩ T. It is no loss to assume that R = S and moreover that S ∩ T is a maximal Sylow intersection in G.

Now Z(S) ≤ C(S ∩ T). So we can find a Sylow 2-subgroup X of G which centralizes S ∩ T and contains Z(S). Then S ∩ T ≤ X, since X normalizes S ∩ T. It follows that S ∩ T ≤ S ∩ X. If S = X we are done. So assume that S ≠ X. Then S ∩ T = S ∩ X, as S ∩ T is a maximal Sylow intersection. In particular Z(S) ≤ S ∩ T. But S ∩ T ≤ Z(X) and |Z(S)| = |Z(X)|. So Z(S) = S ∩ T = Z(X).

Here is our main result:

Theorem 3.4. Let G be a CI-group. Then β(e_{D}, g) = 1_{F}, for each real 2regular element g ∈ G which has defect group D. In particular, the statements (a), (b) and (c) of Theorem 3.2 are equivalent.

Proof. The implications (c) ⟹ (b) ⟹ (a) follow as in Theorem 4.8 of [4].

Let g be a real 2-regular element of G which has defect pair (D, E). Choose s ∈ E\D and set t := sg. Then s and t are 2-elements which invert g and s^{2} = t^{2} lies in D. Let S be a Sylow 2-subgroup of G which contains E. Then t ∈ N(S ∩S^{g}) since (S ∩S^{g})^{t} = S^{st} ∩S^{sgt} = S^{g} ∩S^{t^{2}} = S^{g} ∩S. So ⟨S ∩ S^{g}, t⟩ is a 2-group which contains S ∩S^{g}. We deduce from Lemma 3.3 that t centralizes S ∩ S^{g}. Also s ∈ S also centralizes S ∩ S^{g}, again using Lemma 3.3. So S ∩ S^{g} is a 2-subgroup of C(g). It follows that S ∩ S^{g} = D, as D ≤ S ∩ S^{g} and D is a Sylow 2-subgroup of C(g). The first statement and the implication (a) ⟹ (c) now follow as in Theorem 3.2.

We can now prove:

Proposition 3.5. Let G be a CI-group and let D be a maximal Sylow intersection in G. Then G has a real 2-block with defect group D.

Proof. Note that D is the largest normal 2-subgroup of N(D), and also that it is not a Sylow 2-subgroup of N(D).

We claim that N(D) has a nonidentity real 2-regular element. For suppose otherwise. Then N(D)/D has no nonidentity real 2-regular elements. It follows from the Baer-Suzuki theorem that N(D)/D has a nontrivial normal 2-subgroup, which contradicts the first paragraph.

Theorem 3.4 now shows that N(D) has a real 2-block b with non-maximal defect. But b has a defect group which contains D, by a theorem of R. Brauer. It follows that D is a defect group of b. The proposition now follows from Brauer’s first main theorem.

Theorems 1 and 2 of [6] are consequences of the following corollaries:

Corollary 3.6. Let G be a finite group. Then G has a normal Sylow 2subgroup if and only if G is a CI-group with no real non-principal 2-blocks.

Proof. The ‘only if’ part is straightforward.

Suppose that G is a CI-group which has no real non-principal 2-blocks. Proposition 3.5 implies that G has no maximal Sylow intersections. So G has a normal Sylow 2-subgroup.

We call G a TI-group if every pair of distinct Sylow 2-subgroups of G intersect in the identity.

Corollary 3.7. Let G be a finite group. Then G is a TI-group if and only if G is a CI-group and all real non-principal 2-blocks of G have defect 0.

Proof. The ‘only if’ part is straightforward.

Suppose that G is a CI-group and all real non-principal 2-blocks of G have defect 0. We may assume that G does not have a normal Sylow 2-subgroup. Let D be a maximal Sylow intersection in G. Then G has a real 2-block with defect group D, by Proposition 3.5. It follows from the hypothesis that D = {1}. So G is a TI-group.

4. Extended defect groups for real 2-Blocks

In this section we introduce the notion of defect pairs for real 2-blocks. We defined the relation ⪯ in (1.1). Now ⪯ is almost a partial order, in the sense that if 𝒦 and ℒ are properly real classes and if ℳ is a real class, then

𝒦 ⪯ ℒ and ℒ ⪯ ℳ ⟹ 𝒦 ⪯ ℳ.

Also

(4.1) 𝒦 ⪯ ℒ and ℒ ⪯ 𝒦 ⟹ 𝒦 and ℒ have the same defect pairs.

Set [𝒦] := 𝒦 ∪ 𝒦^{o}, for each class 𝒦 of G, and let

Z^{∗} := ∑ F [𝒦]^{+},

where 𝒦 ranges over the classes of G. Then Z^{∗} is a subalgebra of Z, as it coincides with the set of fixed points of the involutary automorphism x → x^{o} of Z. Each real 2-block idempotent of F G lies in Z^{∗}. By inspecting the proof of Theorem 2.1 of [3], we see that the following is true:

Proposition 4.2. Suppose that ℒ is a real class of G and that 𝒦^{+} is a properly real class which lies in the ideal of Z^{∗} generated by ℒ^{+}. Then 𝒦 ⪯ ℒ.

R. Gow showed in [3, 1.2] that if B ↔ e ↔ ω is a real 2-block of G, then there exists a real 2-regular class 𝒦 of G such that β(e, 𝒦^{+}) ≠ 0 and ω(𝒦^{+}) ≠ 0. He called any such class a real defect class for B. We will call the defect pairs of the real defect classes of B the defect pairs of B.

Proof of Theorem 4.3. Suppose that ω(𝒦^{+}) ≠ 0. Then e = ω(𝒦^{+})^{−1}ϵ(𝒦^{+})e. Also ϵ(𝒦^{+}) = (𝒦^{+})^{2^{n}}, for some integer n > 0, as in Section 2. So e lies in the ideal of Z^{∗} which is generated by 𝒦^{+}. But β(e, ℒ^{+}) ≠ 0. So ℒ ⪯ 𝒦, by Proposition 4.2.

Suppose that β(e, 𝒦^{+}) ≠ 0. Then, using the fact that ω(ℒ^{+}) ≠ 0, the argument of the previous paragraph shows that 𝒦 ⪯ ℒ.

Let P, Q, R and S be subgroups of G. We say that the pairs (P, Q) and (R, S) are conjugate in G if there exists g ∈ G such that R = P^{g} and S = Q^{g}. Our corollary is an immediate consequence of (4.1) and Theorem 4.3:

Corollary 4.4. The defect pairs of a real 2-block are conjugate in G.

5. The number of real 2-blocks with a given defect pair

We begin this section with a result which associates a certain S-orbit of 2-groups to each self-dual (S, S)-double coset.

Lemma 5.1. Suppose that x ∈ G\S and that SxS = Sx^{−1}S. Set (S ∩S^{g})^{∗} := (S ∩ S^{g}) ∪ (Sg ∩ g^{−1}S), for each g ∈ SxS. Then (S ∩ S^{g})^{∗} is a 2-subgroup of G which contains S ∩ S^{g} as a subgroup of index 2. Moroever the (S ∩ S^{g})^{∗} forms a single S-conjugation orbit, and Sg ∩ g^{−1}S coincides with the set {y ∈ Sg | y^{2} ∈ S ∩ S^{g}}.

Proof. First we show that Sg ∩ g^{−1}S is nonempty. We may write g^{−1} = sgt, for certain s, t ∈ S. Then sg = g^{−1}t is an element of Sg ∩ g^{−1}S.

We claim that Sg ∩ g^{−1}S is a right (S ∩ S^{g})-coset. Let a = b^{g} ∈ S ∩ S^{g}, where a, b ∈ S, and let cg = g^{−1}d ∈ Sg ∩ g^{−1}S, where c, d ∈ S. Then (cg)a also lies in Sg ∩ g^{−1}S since g^{−1}(da) = (cg)a = (cb)g. Also (cg)(sg)^{−1} lies in S ∩ S^{g} since cs^{−1} = (cg)(g^{−1}s^{−1}) = (g^{−1}d)(t^{−1}g) = (dt^{−1})^{g}. This proves our claim.

Now sg ∈ N(S ∩ S^{g}), since (S ∩ S^{g})^{sg} = S^{sg} ∩ S^{t} = S^{g} ∩ S. It follows from this and the previous paragraphs that (S ∩ S^{g})^{∗} is a subgroup of G, which contains S ∩ S^{g} as a subgroup of index 2.

Write g = uxv, where u, v ∈ S. Then it is clear that (S ∩ S^{g})^{∗} = (S ∩ S^{x})^{∗v}. So the 2-groups (S ∩ S^{g})^{∗} form a single S-orbit of subgroups of G.

Finally, suppose that y = zg, for z ∈ S. If y^{2} ∈ S∩S^{g}, then y^{2} = u^{g}, for some u ∈ S. Thus y = (g^{−1}ug)(g^{−1}z) = g^{−1}uz lies in Sg∩g^{−1}S. Conversely, suppose that y ∈ Sg ∩ g^{−1}S. Then y = g^{−1}v, for some v ∈ S. Hence zv = y^{2} = (vz)^{g} lies in S ∩ S^{g}. This proves the last statement of the lemma.

The subgroups (S ∩ S^{g})^{∗} have appeared in the literature on self-inverse double cosets. See for example 12.13.(ii) of [2].

Let 𝒦 and ℒ be 2-regular conjugacy classes of G, which have defect groups D and Q respectively. Let x be any element of G. The 2-group S ∩ S^{x} acts by conjugation on 𝒦 ∩ Sx and on ℒ ∩ Sx. So |𝒦 ∩ Sx| ≡ |C(𝒦, x)| (mod2), where C(𝒦, x) := 𝒦 ∩ Sx ∩ C(S ∩ S^{x}). Let O(ℒ, x) denote the set of those orbits of S ∩ S^{x} on ℒ ∩ Sx which have a representative l such that S ∩ S^{x} contains a Sylow 2-subgroup of C(l). Now |ℒ ∩ Sx| = ∑ |S ∩ S^{x} : C_{S∩S^{x}}(l)|. So (|Q| |ℒ ∩ Sx|/|S ∩ S^{x}|) is an integer which has the same parity as |O(ℒ, x)|.

We will use ∑_{x}^{Q<D} denote a sum which ranges over those double cosets SxS for which Q ≤_{G} S ∩ S^{x} ≤_{G} D, and ∑_{x≡x^{−1}}^{Q<D} to denote the restriction of this sum to the self-dual double cosets. These sums are empty unless Q ≤_{G} D. It follows from [12, 1.3.3 and 1.3.4], and the fact that Char(F) = 2, that

β(ϵ(𝒦^{+}), ℒ^{+}) = ∑_{x}^{Q<D} |C(𝒦, x)| |O(ℒ, x)| 1_{F} . (5.2)

Let C^{∗}(𝒦, x) denote the set of elements of C(𝒦, x) which are inverted by some element of (S ∩S^{x})^{∗}, and let O^{∗}(ℒ, x) denote the set of orbits in O(ℒ, x) whose elements are inverted by some element of (S ∩ S^{x})^{∗}.

Proposition 5.3. Suppose that 𝒦 and ℒ are real 2-regular classes of G, with defect groups D and Q respectively. Then

β(ϵ(𝒦^{+}), ℒ^{+}) = ∑_{x≡x^{−1}}^{Q<D} |C^{∗}(𝒦, x)| |O^{∗}(ℒ, x)| 1_{F} .

Proof. By pairing each double coset in (5.2) with its dual, as in the proof of Proposition 2.5, we see that

β(ϵ(𝒦^{+}), ℒ^{+}) = ∑_{x≡x^{−1}}^{Q<D} |C(𝒦, x)| |O(ℒ, x)| 1_{F} .

Suppose that SxS = Sx^{−1}S, where x ∈ G. Let sx ∈ Sx and tx = x^{−1}u ∈ Sx ∩ x^{−1}S, where s, t, u ∈ S. Then (sx)^{−tx} = (u^{−1}x)(x^{−1}s^{−1})(tx) = u^{−1}s^{−1}tx also lies in Sx. Set

y · z := \{ \begin{matrix}y^{z}, & if z ∈ S ∩ S^{x}; \\ (y^{−1})^{z}, & if z ∈ (S ∩ S^{x})^{∗}\(S ∩ S^{x}). \end{matrix}

for each y ∈ Sx and z ∈ (S ∩ S^{x})^{∗}. It is straightforward to show that this defines an action of the 2-group (S ∩ S^{x})^{∗} on Sx.

Now (S ∩ S^{x})^{∗} stabilizes C(𝒦, x), and also each S ∩ S^{x}-orbit in O(ℒ, x). Since (S ∩ S^{x})^{∗} is a 2-group, this implies that

|C(𝒦, x)| ≡ |C^{∗}(𝒦, x)| (mod2) and |O(ℒ, x)| ≡ |O^{∗}(ℒ, x)| (mod2).

The proposition follows from this.

Proof of Theorem 5.2. Recall the notation established in Section 1.

Let B_{1} ↔ e_{1} ↔ ω_{1}, …, B_{u} ↔ e_{u} ↔ ω_{u}, be a complete list of the (real) 2-blocks of G which have defect pair (D, E). Suppose that 1 ≤ i, j ≤ u. Then

δ_{ij} = ω_{j}(e_{i}) = ∑ β(e_{i}, 𝒦^{+})ω_{j}(𝒦^{+}), (5.5)

where 𝒦 runs through the conjugacy classes of G. Suppose that 𝒦 ≠ 𝒦^{o}. Then the contribution of 𝒦 and 𝒦^{o} to (5.5) is

β(e_{j}, 𝒦^{+})ω_{i}(𝒦^{+}) + β(e_{j}, 𝒦^{o+})ω_{i}(𝒦^{o+}) = 2 · β(e_{j}, 𝒦^{o+})ω_{i}(𝒦^{o+}) = 0.

Also any real class which occurs with non-zero multiplicity in (5.5) is 2-regular and is not the trivial class. So any such class is properly real. It follows from Theorem 4.3 that

δ_{ij} = ∑_{k=1}^{v} β(e_{i}, 𝒦_{k}^{+})ω_{j}(𝒦_{k}^{+}). (5.6)

Form the u × v-matrices A and B by setting the i, j-th entry of A as A_{ij} = β(e_{i}, 𝒦_{j}^{+}) and the i, j-th entry of B as B_{ij} = ω_{i}(𝒦_{j}^{+}). Then AB^{T} is the u × u identity matrix, by (5.6). It follows that the v × v-matrix B^{T} A has rank u.

Suppose that B ↔ e ↔ ω is a non-real 2-block of G and that 1 ≤ i, j ≤ v. Then, since 𝒦_{i} = 𝒦_{i}^{o} and 𝒦_{j} = 𝒦_{j}^{o}, the contribution of e and e^{o} to β(ϵ(𝒦_{i}^{+}), 𝒦_{j}^{+}) is

ω(𝒦_{i}^{+})β(e, 𝒦_{j}^{+}) + ω^{o}(𝒦_{i}^{+})β(e^{o}, 𝒦_{j}^{+}) = 2 · ω(𝒦_{i}^{+})β(e, 𝒦_{j}^{+}) = 0.

Thus

β(ϵ(𝒦_{i}^{+}), 𝒦_{j}^{+}) = β (∑ ω(𝒦_{i}^{+})e, 𝒦_{j}^{+}) = ∑ ω(𝒦_{i}^{+})β(e, 𝒦_{j}^{+}),

where B ↔ e ↔ ω runs through the real 2-blocks of G. So by Theorem 4.3 we have

β(ϵ(𝒦_{i}^{+}), 𝒦_{j}^{+}) = ∑_{k=1}^{u} ω_{k}(𝒦_{i}^{+})β(e_{k}, 𝒦_{j}^{+}).

The sum on the right hand side is the i, j-th entry of the matrix B^{T} A. We conclude that the v × v matrix M with i, j-th entry M_{ij} = β(ϵ(𝒦_{i}^{+}), 𝒦_{j}^{+}) has rank u.

It now follows from Proposition 5.3 that u = 0 if w = 0, and

M_{i,j} = ∑_{k=0}^{w} C^{∗}(𝒦_{i}, x_{k})C^{∗}(𝒦_{j}, x_{k}),

if w > 0. But C^{∗}(𝒦_{i}, x_{k}) = N_{ik} and C^{∗}(𝒦_{j}, x_{k}) = N_{jk}. We conclude that M = N · N^{T}, which completes the proof.

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