Abstract
Knörr 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.
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1 Introduction
Let F be an algebraically closed field of characteristic \(p>0\), and let Q be a normal p-subgroup of a finite group G. Motivated by Alperin’s weight conjecture, Knörr [3] proved that the number of blocks of defect zero in the group algebra F[G/Q] can be computed as the dimension of a certain ideal in the center \({\mathrm Z}(FG)\).
Knörr’s approach used the permutation FG-module on the cosets of a Sylow p-subgroup P of G and its endomorphism ring. Here we present a different approach to his result by making use of properties of symmetric algebras. Applied to group algebras, our main result is as follows:
Theorem 1.1
In the situation described above, the number of blocks of defect zero in F[G/Q] coincides with the dimension of the ideal
of \({\mathrm Z}(FG)\).
Let us explain the notation used in Theorem 1.1. 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 \(X^+ := \sum _{x \in X} x \in FG\). Then \(FG \cdot G_p^+ = G_p^+ \cdot FG\) is a principal (two-sided) ideal of FG contained in the (left and right) socle \({\mathrm S}(FG)\). By a result of Tsushima [8] (see also [4]), the annihilator of \(G_p^+\) in FG is the sum of the radical \({\mathrm J}(FG)\) of FG and the left ideals FGe of FG where e ranges over the primitive idempotents in FG such that \(\dim FGe\) is divisible by p|P|.
For any subgroup X of G,
is a subalgebra of FG and, for any subgroup Y of X,
is the relative trace (transfer) map. Observe that in Theorem 1.1, we have \(FG \cdot G_p^+ \subseteq (FG)^Q\) since \(uG_p^+ = G_p^+\) for all \(u \in Q\). Since \(FG \cdot G_p^+\) is an ideal in \((FG)^Q\), standard properties of the transfer map imply that \({\mathrm W}_Q(FG):= \textrm{Tr}_Q^G(FG \cdot G_p^+)\) is an ideal in \((FG)^G = {\mathrm Z} (FG)\).
Observe also that, by definition, we have \({\mathrm W}_Q(FG) = 0\) unless \(Q = {\mathrm O}_p(G)\); in fact, note that \(FG \cdot G_p^+\) is contained in \((FG)^N\) where \(N:= {\mathrm O}_p(G)\) by the argument above. In case \(Q \ne N\), this implies \(\textrm{Tr}_Q^N(x) = |N:Q|x = 0\) and thus \(\textrm{Tr}_Q^G(x) = 0\) for every \(x \in FG \cdot G_p^+\).
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 \(\mathbb {Z}/ p\mathbb {Z}\); 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 2, we first recall some properties of symmetric algebras and then prove a version of Knörr’s result in this context. In Section 3, we apply the results of Section 2 to group algebras, prove Theorem 1.1, and finish with some additional remarks.
2 Symmetric algebras
Let F be an algebraically closed field, and let A be a symmetric F-algebra with symmetrizing linear form \(\lambda {:}\,A \longrightarrow F\). Moreover, let I be an ideal of A with the property that the F-algebra A/I is also symmetric, and let \(\mu {:}\,A/I \longrightarrow F\) be a corresponding symmetrizing linear form. Then there exists a unique element z in the center \(\textrm{Z}(A)\) of A such that \(\mu (a+I) = \lambda (az)\) for all \(a \in A\). Consequently, I is the annihilator of z in A. We denote by \(\nu {:}\,A \longrightarrow A/I\), \(a \longmapsto a+I\), the canonical epimorphism, and by \(\nu ^*{:}\,A/I \longrightarrow A\) the adjoint of \(\nu \) (with respect to \(\lambda \) and \(\mu \)). Thus
Explicitly, we have
in particular, \(\nu ^*\) is a monomorphism of A-A-bimodules. This implies that \(\nu ^*(\textrm{J}(A/I)) \subseteq \textrm{J}(A)\), \(\nu ^*(\textrm{S}(A/I)) \subseteq \textrm{S}(A)\), and \(\nu ^*(\textrm{Z}(A/I)) = \textrm{Z}(A) \cap Az \subseteq \textrm{Z}(A)\); here \(\textrm{J}(A)\) denotes the (Jacobson) radical of A, and \(\textrm{S}(A)\) denotes the (left and right) socle of A (see [1]). We also conclude:
Proposition 2.1
In the situation above, let L be an ideal of \(\textrm{Z}(A/I)\). Then \(\nu ^*(L)\) is an ideal of \(\textrm{Z}(A)\).
Proof
This follows since L is a \(\textrm{Z}(A)\)-module and since \(\nu ^*\) is a monomorphism of \(\textrm{Z}(A)\)-modules. \(\square \)
In particular, we have \(\nu ^*(\textrm{R}(A/I)) \subseteq \textrm{R}(A)\) where \( \textrm{R}(A) := \textrm{Z}(A) \cap \textrm{S}(A)\) denotes the Reynolds ideal, an ideal in \(\textrm{Z}(A)\) (cf. [1, 6]). Similarly, \(\nu ^*(\textrm{H}(A/I))\) and \(\nu ^*(\textrm{Z}_0(A/I))\) are ideals of \(\textrm{Z}(A)\); here \(\textrm{H}(A)\) denotes the Higman ideal, an ideal in \(\textrm{Z}(A)\), and \(\textrm{Z}_0(A)\) is the sum of the centers of the blocks of A which are simple F-algebras, also an ideal of \(\textrm{Z}(A)\). Recall that \(\textrm{Z}_0(A) \subseteq \textrm{H}(A) \subseteq \textrm{R} (A)\) and \(\textrm{R}(A)^2 = \textrm{Z}_0(A)\), so that also \(\textrm{H}(A)^2 = \textrm{Z}_0(A)\); see [2, Section 4].
We obtain the following version of Knörr’s theorem in the context of symmetric algebras:
Corollary 2.2
In the situation above, the dimension of the ideal \(\nu ^*(\textrm{Z}_0(A/I))\) of \(\textrm{Z}(A)\) is the number of blocks of A/I which are simple F-algebras.
The inclusion \(\textrm{Z}_0(A) \subseteq \textrm{H}(A) \subseteq \textrm{R}(A)\) (cf. [2]) implies that, in Corollary 2.2, we have
3 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 \(\nu : FG \longrightarrow F[G/Q]\) by I. Then \(I = FG \cdot \textrm{J} (FQ) = \textrm{J}(FQ) \cdot FG\), and we identify FG/I and F[G/Q].
The group algebra FG is a symmetric F-algebra with symmetrizing linear form
We denote the analogous symmetrizing linear form of F[G/Q] by \(\mu \). Then, in the notation of Section 2, we have \(z = Q^+\).
It is well-known that \(\textrm{H}(FG) = \textrm{Tr}_1^G(FG)\) is the ideal of \((FG)^G = \textrm{Z}(FG)\) which, as a vector space over F, is spanned by the class sums of the conjugacy classes of p-defect zero in G. Furthermore, the following holds:
in fact, as noted in Section 2, we have \(\textrm{H}(FG) \cdot G_p^+ \subseteq \textrm{R}(FG)^2 = \textrm{Z}_0(FG)\), and the annihilator of \(G_p^+\) in \(\textrm{Z}(FG)\) is \(\textrm{J}(\textrm{Z}(FG))\). We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1
Recall first that a block of FG is a simple F-algebra if and only if it has defect zero. Hence, by Corollary 2.2, the number of blocks of defect zero in F[G/Q] is the dimension of the ideal \(\nu ^*(\textrm{Z}_0(F[G/Q])) = \nu ^*(\textrm{H}(F[G/Q]) \cdot (G/Q)_p^+)\) of \(\textrm{Z} (FG)\). Let T be a set of representatives (in G) for the cosets of p-elements in G/Q. Then \(\nu (T^+) = (G/Q)_p^+\) and \(Q^+T^+ = G_p^+\). Moreover, we have \(\nu ^*(gQ) = gQ^+\) and
for \(g \in G\). Thus we obtain
and Theorem 1.1 follows. \(\square \)
A short computation shows:
for \(g \in G\). Thus we have \(\textrm{Tr}_Q^G(gG_p^+) = 0\) unless \(\textrm{C}_Q(g)\) is a Sylow p-subgroup of \(\textrm{C}_G(g)\).
Similarly, if \(0 \ne \textrm{Tr}_Q^G(g G_p^+) = \nu ^*(\textrm{Tr}_1^{G/Q}(gQ) \cdot (G/Q)_p^+)\), then we have \(0 \ne \textrm{Tr}_1^{G/Q}(gQ) \cdot (G/Q)_p^+\). Thus [5, Lemma N] implies that there exists a Sylow p-subgroup P of G such that \(P \cap gPg^{-1} = Q\).
We denote by \(g_p\) and \(g_{p'}\) the p-factor and the \(p'\)-factor of g. If \(\textrm{C}_Q(g)\) is a Sylow p-subgroup of \(\textrm{C}_G(g)\), then \(g_p \in \textrm{C}_G(g)_p = \textrm{C}_Q(g)\) and therefore \(g_pG_p^+ = G_p^+\). Consequently, we have \(\textrm{Tr}_Q^G(gG_p^+) = \textrm{Tr}_Q^G(g_{p'} \cdot G_p^+)\). This shows that the vector space \(\textrm{Tr}_Q^G(FG \cdot G_p^+)\) is spanned by the elements \( \textrm{Tr}_Q^G(gG_p^+)\) where g ranges over the \(p'\)-elements in G such that \(\textrm{C}_Q(g)\) is a Sylow p-subgroup of \(\textrm{C}_G(g)\) and such that there exists a Sylow p-subgroup P of G such that \(P \cap gPg^{-1} = Q\).
On the other hand, we have
It is well-known that \((FG)_Q^G\) is an ideal in \(\textrm{Z}(FG)\) 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. Recall also that, as observed at the end of Section 2, \(\textrm{Tr}_Q^G(FG \cdot G_p^+)\) is contained in the Reynolds ideal \(\textrm{R}(FG)\).
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Acknowledgements
The author is grateful to G.R. Robinson for discussions which led to the present paper, and to the referee for a careful reading and several helpful suggestions.
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Dedicated to Geoff Robinson, on the occasion of his 70th birthday.
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Külshammer, B. On a theorem of Knörr. Arch. Math. 122, 385–389 (2024). https://doi.org/10.1007/s00013-023-01961-z
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DOI: https://doi.org/10.1007/s00013-023-01961-z