ON A CERTAIN IDEAL OF KÜLSHAMMER IN THE CENTRE OF A GROUP ALGEBRA

JOHN MURRAY

Abstract. Let G be a finite group and let F be a splitting field of characteristic p > 0. We show that I^{2} = E_{0}, where I is a certain ideal of the centre Z of F G, and E_{0} is the span of the block idempotents of defect zero.

Let G be a finite group and let F be a field of characteristic p. We shall assume that F is a splitting field for G. Let λ denote the linear map F G → F, given by

λ(∑_{g∈G} a_{g}g) = a_{1},

for ∑_{g∈G} a_{g}g ∈ F G. The map B : F G × F G → F,

B(a, b) = λ(ab), for a, b ∈ F G,

is a nondegenerate symmetric bilinear form on F G. In fact B is associative in the sense that

B(ab, c) = B(a, bc), for a, b, c ∈ F G.

If V is an F -subspace of F G, then V^{⊥} will denote the dual space

V^{⊥} := {a ∈ F G | B(a, b) = 0, for all b ∈ V }.

If A is a subalgebra of F G, and V is a right A-module, then V^{⊥} is a left A-module. Let K := Z^{⊥}, where Z is the centre of F G. Then by the above remarks, K is a Z-submodule of F G. It is clear that

(1) K = {∑_{g∈G} a_{g}g | ∑_{g∈K} a_{g} = 0, for all conjugacy classes K of G},

as the class sums K^{+} := ∑_{g∈K} g form an F -basis for Z. The following lemma is due to R. Brauer [B56]:

Lemma 2.

\begin{aligned}a ∈ K & ⟹ & a^{p} ∈ K, \\ (a + b)^{p} & ≡ & a^{p} + b^{p} mod K, \end{aligned}

for all a, b ∈ F G.

Set

N := \{ \begin{matrix}4, & if p = 2, \\ p, & if p is an odd prime, \end{matrix}

and define

T := {x ∈ F G | x^{N} ∈ K}.

Date: Submitted February 1, Revised March 14, 2000.

1991 Mathematics Subject Classification. 20C20.

Then Lemma 2 implies that T is a subspace of F G which contains K. An easy argument shows that T is a Z-submodule of F G. Since T ⊇ K, it follows that

I := T^{⊥}

is an ideal of Z.

For each conjugacy class K, set

Ω(K) := {g ∈ G | g^{N} ∈ K}.

So Ω(K) is a union of conjugacy classes of G. We have the following (see (38) of [K91]):

Lemma 3.

T = {∑_{g∈G} a_{g}g | ∑_{g∈Ω(K)} a_{g} = 0, for all classes K}.

Thus {Ω(K)^{+} | K a conjugacy class of G, Ω(K) ≠ ϕ} forms a basis for I.

Proof. Say a = ∑_{g∈G} a_{g}g ∈ F G. Then

a ∈ T ⟺ (∑_{g∈G} a_{g}g)^{N} ∈ K, by definition of T

⟺ ∑_{g∈G} a_{g}^{N}g^{N} ∈ K, using Lemma 2

⟺ ∑_{g^{N} ∈K} a_{g}^{N} = 0, for all classes K, by (1)

⟺ ∑_{g∈Ω(K)} a_{g} = 0, for all classes K, as F has characteristic p.

The last statement now follows from the first.

Proposition 4. Let z ∈ Z and suppose that z^{N} = 0. Then Iz = zI = 0.

Proof. Let i ∈ I and x ∈ F G. It follows from the hypothesis that zx ∈ T. Thus B(iz, x) = B(i, zx) = 0. Since x ∈ F G was arbitrary, the nondegeneracy of λ implies that iz = 0.

Let Z_{0} denote the F -subspace of Z spanned by the class sums of p-defect zero, and let E_{0} denote the F -subspace of Z_{0} spanned by the block idempotents of defect zero. Then Z_{0} is an ideal of Z, using an argument due to R. Brauer. Moreover Iizuka and Watanabe [IW73] have shown that

(Z_{0})^{2} = E_{0}, (5)

and

Z_{0} J(F G) = 0, (6)

where J(F G) is the Jacobson radical of F G. See (1.E) of [O80] also. A proof of the following result was indicated in [M99]:

Lemma 7. I^{2} ⊆ Z_{0}.

Proof. Let K, L and M be classes of G. The coefficient of M^{+} in Ω(K)^{+}Ω(L)^{+} is given as the cardinality, modulo p, of the set

Φ(M) := {(k, l) ∈ Ω(K) × Ω(L) | kl = m},

where m is a fixed element of M. Let D be a defect group of m. Then D acts by conjugation on the pairs in Φ(M). So |Φ(M)| ≡ |Φ_{D}(M)| mod p, where

Φ_{D}(M) := {(k, l) ∈ (C(D) ∩ Ω(K)) × (C(D) ∩ Ω(L)) | kl = m}.

Now let Ω(Z(D)) be the subgroup of Z(D) consisting of all z ∈ Z(D) such that z^{N} = 1. Then Ω(Z(D)) acts freely on Φ_{D}(M) via

(k, l) → (kz, z^{−1}l), for (k, l) ∈ Φ_{D}(K), and z ∈ Ω(Z(D)).

It follows that |Φ_{D}(M)| ≡ 0 mod p unless Ω(Z(D)) = {1} ⟺ D = {1} i.e. unless M has p-defect zero. The lemma follows.

Corollary 8. I(I ∩ J(F G)) = 0 and hence (I ∩ J(F G))^{2} = 0.

Proof. Suppose that j ∈ I ∩ J(F G). Then j ∈ J(F G), as I ⊆ Z. Also j^{2} ∈ Z_{0}, by Lemma 7. So j^{3} = j(j^{2}) = 0, using (6). But then j^{N} = j^{N−3}j^{3} = 0. Proposition 4 now implies that I(I ∩ J(F G)) = 0. The equality (I ∩ J(F G))^{2} = 0 follows immediately.

We can now prove our main result.

Theorem 9. I^{2} = E_{0}.

Proof. Let E denote the F -subspace of Z spanned by the block idempotents. Then

Z = E ⊕ J,

as F -algebras, where J = Z ∩ J(F G) is the Jacobson radical of Z. Now J is nil and the map x → x^{p} is an automorphism of F. It follows that there exists m ≥ 0 such that e^{p^{m}} = e and j^{p^{m}} = 0, for all e ∈ E and j ∈ J.

If i_{1}, i_{2} ∈ I, write

i_{k} = e_{k} + j_{k}, (k = 1, 2),

where e_{k} ∈ E and j_{k} ∈ J. Then e_{k} = e_{k}^{p^{m}} + j_{k}^{p^{m}} = i_{k}^{p^{m}} ∈ I. It follows that e_{k} ∈ I and j_{k} ∈ I ∩ J(F G). So

i_{1}i_{2} = e_{1}e_{2} + e_{1}j_{2} + j_{1}e_{2} + j_{1}j_{2} = e_{1}e_{2},

using Corollary 8. Thus I^{2} ⊆ E ∩ Z_{0} = E_{0}, using Lemma 7.

The oppositite inequality I^{2} ⊇ E_{0} follows from I ⊇ Z_{0} and (5).

We also have:

Proposition 10. Let K be a p-singular class of G. Then Ω(K)^{+} ∈ J(F G). In particular, Ω(K)^{+}Ω(L)^{+} = 0, for each class L of G.

Proof. Let B be a p-block of G, with associated central character ω. If B has positive defect, then ω((Ω(K)^{+})^{2}) = 0, using Lemma 7, and so ω(Ω(K)^{+}) = 0. On the other hand, if B has defect zero, then ω(Ω(K)^{+}) = 0, as Ω(K) is a union of p-singular classes. We deduce that Ω(K)^{+} ∈ J(F G). The last statement now follows from Corollary 8.

If g ∈ G, we may write g = g_{p}g_{p′} = g_{p′}g_{p}, for a unique p-element g_{p} and a unique p-regular element g_{p′}. We call g_{p} the p-part of g and g_{p′} the p-regular part of g. Let K be a p-regular class of G. The p-regular section S(K) of G which contains K is defined as

S(K) := {g ∈ G | g_{p′} ∈ K}.

Setting L^{N} = {g^{N} | g ∈ L}, for each class L of G, we note that

S(K) = ⋃_{L⊂S(K)} Ω(L^{N}).

The p-regular section sums S(K)^{+} span an ideal R of Z, known as Reynolds Ideal. We have the following chain of ideals of Z:

E_{0} ⊆ Z_{0} ⊆ R ⊆ I.

Now R = J(F G)^{⊥} ∩ Z, by (39) of [K91]. It follows easily that

R^{2} = E_{0}.

So Theorem 9 is an improvement on this fact.

Corollary 11. Suppose that K, L are p-regular classes of G. Then

S(K)^{+}S(L)^{+} = Ω(K^{N})^{+}Ω(L^{N})^{+}.

Proof. This follows from Proposition 10, and the fact that K and L are the only p-regular classes in S(K) and S(L), respectively.

The following extends results in [IW73] and [M99]:

Corollary 12. G has a p-block of defect zero if and only if there exists p-regular classes K, L of G such that Ω(K)^{+}Ω(L)^{+} ≠ 0, i.e. there exists g ∈ G (necessarily of p-defect zero) such that the cardinality of the set

{(x, y) ∈ G × G | x^{N} ∈ K, y^{N} ∈ L, xy = g}

is nonzero modulo p.

We now give some examples in the exceptional case where p = 2 and F has characteristic 2. Set

T_{1} := {x ∈ F G | x^{2} ∈ K}.

Then

I_{1} := T_{1}^{⊥}

is an ideal of Z, and has as F -basis {Ω_{1}(K)^{+}}, where K ranges over the conjugacy classes of G, and

Ω_{1}(K) := {g ∈ G | g^{2} ∈ K}.

Although E_{0} ⊆ I_{1}^{2} ⊆ Z_{0}, and our results can be extended to show that E_{0} = I_{1}^{3}, it is not generally true that E_{0} = (I_{1})^{2}. For instance, if G = 𝒮_{7}, the symmetric group on 7-symbols, then 0 = E_{0} ⊂ I_{1}^{2} = Z_{0}, while if G = M_{23}, the Mathieu group of degree 23, then 0 ⊂ E_{0} ⊂ I_{1}^{2} ⊂ Z_{0}. On the other hand, if G = 𝒮_{3}, then E_{0} = I_{1}^{2} = Z_{0}.

Let R denote the set of elements of G which have 2-defect zero and which are conjugate to their inverses. We showed in [M99] that

(Ω_{1}(1_{G})^{+})^{2} = R^{+}.

It follows that R^{+}Z ⊆ I_{1}^{2}. We have not found an example where R^{+}Z ≠ I_{1}^{2}.

References

[B56] Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung Math. Zeit. 63 (1956), 406–444.

[IW73] K. Iizuka, 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.

[K91] Burkhard Külshammer, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Progress in Math. 95 (1991), 425-442.

[M99] John Murray, Blocks of defect zero and products of elements of order p, J. Algebra 214 (1999), 385–399.

[O80] T. Okuyama, Some studies on group algebras, Hokkaido Mathematical J. 9 (1980), 217-221.

Mathematics Department, University College Dublin, Belfield, Dublin 4, Ireland

   E-mail address: jcmurray@eircom.ie