On a Theorem of Kn\"orr

Kn\"orr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his construction to symmetric algebras.

Theorem 1.In the situation described above, the number of blocks of defect zero in F [G/Q] coincides with the dimension of the ideal Let us explain the notation used in Theorem 1.We denote by G p the set of p-elements in G, and by G p ′ the set of p ′ -elements in G.For any subset X of G, we set X + := x∈X x ∈ F G.
Then F G • G + p = G + p • F G is a principal (two-sided) ideal of F G contained in the (left and right) socle S(F G).By a result of Tsushima in [8] (see also [4]), the annihilator of G + p in F G is the sum of the radical J(F G) of F G and the left ideals F Ge of F G where e ranges over the primitive idempotents in F G such that dim F Ge is divisible by p|P |.
For any subgroup X of G, is a subalgebra of F G and, for any subgroup Y of X, is the relative trace (transfer) map.Observe that, in Theorem 1, we have . By a result of Robinson [7], the number of blocks of defect zero of a group algebra can be computed as the rank of a certain matrix with coefficients in Z/pZ; see also [5].In [9], Wang and Zhang have translated Robinson's theorem from G/Q to G.
This paper is organized as follows: In Section 1, we first recall some properties of symmetric algebras and then prove a version of Knörr's result in this context.In Section 2 we apply the results of Section 1 to group algebras, prove Theorem 1 and finish with some additional remarks.

Symmetric algebras
Let F be an algebraically closed field, and let A be a symmetric F -algebra with symmetrizing linear form λ : A −→ F .Moreover, let I be an ideal of A such that the F -algebra A/I is also symmetric, and let µ : A/I −→ F be a corresponding symmetrizing linear form.Then there exists a unique element z in the center Z(A) of A such that µ(a + I) = λ(az) for all a ∈ A. Consequently, I is the annihilator of z in A. We denote by ν : A −→ A/I, a −→ a + I, the canonical epimorphism, and by ν * : A/I −→ A the adjoint of ν (with respect to λ and µ).Thus Explicitly, we have ν * (a + I) = az for all a ∈ A; in particular, ν * is a monomorphism of A-A-bimodules.This implies that ν * (J(A/I)) ⊆ J(A), ν * (S(A/I)) ⊆ S(A) and ν * (Z(A/I)) = Z(A) ∩ Az ⊆ Z(A); here J(A) denotes the (Jacobson) radical of A, and S(A) denotes the (left and right) socle of A; see [1].We also conclude: Proposition 2. In the situation above, let L be an ideal of Z(A/I).Then ν * (L) is an ideal of Z(A).
Proof.This follows since L is a Z(A)-module and since ν * is a monomorphism of Z(A)modules.
In particular, we have ν * (R(A/I)) ⊆ R(A) where R(A) := Z(A) ∩ S(A) denotes the Reynolds ideal, an ideal in Z(A) (cf.[1] or [6]).Similarly, ν * (H(A/I)) and ν * (Z 0 (A/I)) are ideals of Z(A); here H(A) denotes the Higman ideal, an ideal in Z(A), and Z 0 (A) = H(A) 2 is the sum of the blocks of A which are simple F -algebras.Hence Z 0 (A) is also an ideal of Z(A); see [2].We obtain the following version of Knörr's theorem in the context of symmetric algebras: Corollary 3. In the situation above, the dimension of the ideal ν * (Z 0 (A/I)) of Z(A) is the number of blocks of A/I which are simple F -algebras.[2]) implies that, in Corollary 3, we have

Group algebras
In the following, let F be an algebraically closed field of characteristic p > 0, and let Q be a normal p-subgroup of a finite group G.We denote the kernel of the canonical epimorphism The group algebra F G is a symmetric F -algebra with symmetrizing linear form We denote the analogous symmetrizing linear form of F [G/Q] by µ.Then, in the notation of Section 1, we have z which, as a vector space over F , is spanned by the class sums of the conjugacy classes of p-defect zero in G. Furthermore, we have ).We are now in a position to prove Theorem 1.
Proof (of Theorem 1).Recall first that a block of F G is a simple F -algebra if and only if it has defect zero.Hence, by Corollary 3, the number of blocks of defect zero in Thus we obtain: and Theorem 1 follows.
A short computation shows: p .Thus Lemma N in [5] implies that there exists a Sylow p-subgroup P of G such that P ∩ gP g −1 = Q.
We denote by g p and g p ′ the p-factor and the p ′ -factor of g.
. This shows that the vector space Tr G Q (F G • G + p ) is spanned by the elements Tr G Q (gG + p ) where g ranges over the p ′ -elements in G such that C Q (g) is a Sylow p-subgroup of C G (g) and such that there is a Sylow p-subgroup P of G such that P ∩ gP g −1 = Q.
On the other hand, we have It is well-known that (F G) G Q is an ideal in Z(F G) which, as a vector space over F , is spanned by the class sums of the conjugacy classes of G whose defect groups are contained in Q.