Abelian Defect Group Research Articles

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GLn(q) with defect group Cpα×Cpα, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GLn(q) when p=3, α=1, in a different way.

Derived equivalence in 𝑆𝐿₂(𝑝²)

We present a proof that Broué’s Abelian Defect Group Conjecture is true for the principal p p -block of the group S L 2 ( p 2 ) SL_2(p^2) . Okuyama has independently obtained the same result using a different approach.

Twisted group algebras, normal subgroups and derived equivalences

We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broue's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p′-extensions of blocks with cyclic defect groups.

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theory is the relationship between the p-local structure and the p-modular representation theory of a given finite group. In [5], Broué poses some startling conjectures. For example, he conjectures that if e is a p-block of a finite group G with abelian defect group D and if f is the Brauer correspondent block of e of the normalizer, NG(D), of D then e and f have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p. Some evidence for this conjecture has been obtained using an important Morita analog for derived categories of Rickard [11]. This result states that the existence of a tilting complex is a necessary and sufficient condition for the equivalence of two derived categories. In [5], Broué also defines an equivalence on the character level between p-blocks e and f of finite groups G and H that he calls a ‘perfect isometry’ and he demonstrates that it is a consequence of a derived category equivalence between e and f. In [5], Broué also poses a corresponding perfect isometry conjecture between a p-block e of a finite group G with an abelian defect group D and its Brauer correspondent p-block f of NG(D) and presents several examples of this phenomena. Subsequent research has provided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blocks e, f of finite groups G, H there are also perfect isometries between p-blocks of p-local subgroups corresponding to e and f and these isometries are compatible in a precise sense. In [5], Broué calls such a family of compatible perfect isometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defining a p-locally compatible family of derived equivalences. In this important paper, he defines a ‘splendid tilting complex’ for p-blocks e and f of finite groups G and H with a common p-subgroup P. Then he demonstrates that if X is such a splendid tilting complex, if P is a Sylow p-subgroup of G and H and if G and H have the same ‘p-local structure’, then p-local splendid tilting complexes are obtained from X via the Brauer functor and ‘lifting’. Consequently, in this situation, we obtain an isotypy when e and f are the principal blocks of G and H. Linckelmann [9] and Puig [10] have also obtained important results in this area. In this paper, we refine the methods and program of [11] to obtain variants of some of the results of [11] that have wider applicability. Indeed, suppose that the blocks e and f of G and H have a common defect group D. Suppose also that X is a splendid tilting complex for e and f and that the p-local structure of (say) H with respect to D is contained in that of G, then the Brauer functor, lifting and ‘cutting’ by block indempotents applied to X yield local block tilting complexes and consequently an isotypy on the character level. Since the p-local structure containment hypothesis is satisfied, for example, when H is a subgroup of G (as is the case in Broué's conjectures) our results extend the applicability of these ideas and methods.

Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

Letbbe the principalp-block of a finite groupGwith an abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is an elementary abelian 2-group (respectively, a dihedral group of order 8) andp≠3, thenband its Brauer correspondent, considered as blocks ofGandNG(P) are isotypic and, in particular, there is a perfect isometry between them.

On perfect isometries and isotypies in alternating groups

Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broue conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized Fitting subgroup isomorphic to an alternating group. Let G be a finite group and let 0 be a complete discrete valuation ring with field of quotients K of characteristic 0 and residue class field k of characteristic p > 0. We suppose that K contains a primitive IGI-th root of unity. In [4, (6.1)] Michel Broue posed the following Isotypy Conjecture. Let e be a block of OG with abelian defect group D and let f be the Brauer correspondent of e in (9NG (D). Then e and f are isotypic blocks. If G has an abelian Sylow p-subgroup, then the conjecture can be posed for the principal block of 0G. In this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian Sylow p-subgroup (see [10, (5E)]). In this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups. In addition, the extended conjecture is proved for the principal p-block of almost simple groups with abelian Sylow p-subgroups and generalized Fitting subgroup isomorphic to an alternating group. We recall the basic definitions. Let -: 0 k be the canonical quotient mapping and let -: OG kG be the induced 0-algebra homomorphism of the group algebras. In particular, -: e -*e induces a bijection between central idempotents of OG and kG. If e is a block idempotent of OG, let KGeMod be the category of left KGe-modules of finite type and let ZK (G, e) be the Grothendieck group of KGeMod. Let Gv be the set of irreducible characters of G over K. We identify R.K (G, e) with the free abelian group on (G, e)v = {X E Gv X(ge) = X(g) for all g E G}. Let CF(G, K) be the K-space of K-valued class functions on G, and let CF(G, e, K) be the K-subspace of class functions ae in CF(G, K) such that ao(ge) = a(g). The Received by the editors March 26, 1996. 1991 Mathematics Subject Classification. Primary 20C15, 20C20; Secondary 20C30. The first author was supported in part by NSF grant DMS 9100310. The second author was supported in part by NSA grant MDA 904 92-H-3027. ?1997 American Mathematical Society

Perfect Isometries and Isotypies for Blocks with Abelian Defect Groups and the Inertial Quotients Isomorphic toZ3×Z3

Letbbe ap-block of a finite groupGwith abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is isomorphic toZ3×Z3andp≠2,7, then there is a perfect isometry from the group of generalized characters of some twisted group algebra of the semidirect product ofEandPonto the group of generalized characters ofGinb, and, furthermore,band its Brauer correspondent inNG(P) are isotypic.

Perfect Isometries and Isotypies for Blocks with Abelian Defect Groups and the Inertial Quotients Isomorphic toZ4×Z2

Letbbe ap-block of a finite groupGwith abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is isomorphic toZ4×Z2andp≧7, then there is a perfect isometry from the group of generalized characters of some twisted group algebra of the semidirect product ofEandPonto the group of generalized characters ofGinb, and, furthermore,band its Brauer correspondent inNG(P) are isotypic.

Perfect Isometries for Blocks with Abelian Defect Groups and Cyclic Inertial Quotients of Order 4

Let b be a p-block of a finite group G with an abelian defect group P and e a root of b in CG(P). If the inertial quotient E (= NG(P, e)/P · CG(P)) is a cyclic group of order 4, there is a perfect isometry from the group of generalized characters of the group algebra of the semidirect product of E and P onto the group of generalized characters of G in b.

Perfect Isometries for Blocks with Abelian Defect Groups and Dihedral Inertial Quotients of Order 6

Let b be a p-block of a finite group G with an abelian defect group P and let e be a root of b in CG(P). If the inertial quotient E (= NG(P, e)/P · CG(P)) is a dihedral group of order 6, then there is a perfect isometry from the group of generalized characters of the group algebra of the semidirect product of E and P onto the group of generalized characters of G in b.

Blocks of Endomorphism Algebras

Let O G be a p-modular group algebra, let H be a subgroup of G containing the normaliser of a p-subgroup P, let A be an O G-module, and B an O H-module. After defining defect groups of blocks of End O G ( A) and of End O H ( B), we associate a certain block of End( O H)( A ↓ H ) with each block of End O G ( A) having defect group P. Similarly, we associate a certain block of End O G ( B ↑ G ) with each block of End O H ( B) with defect group P. These associations are compatible with the correspondences of Brauer and of Green, and, in particular, they partly generalise Brauer′s First and Second Main Theorems. The theory simplifies when working within blocks of group algebras with abelian defect groups.

Perfect Isometries for Blocks with Abelian Defect Groups and Klein Four Inertial Quotients

Let b be a p-block of a finite group G with an abelian defect group P and e a root of b in C G ( P). If the inertial quotient E (= N G ( P, e)/ C G ( P)) is a Klein four group, there is a so called perfect isometry from the group of generalized characters of a suitable twisted group algebra of the semidirect product of E and P onto the group of generalized characters of G in b.

Notes on p-blocks of characters of finite groups

Let G be a finite group and let p be a fixed prime number. In this paper we shall study the numbers of irreducible characters in a p-block of G with an abelian defect group and the contributions of subsections to the inner product of irreducible characters of G. As is seen in [ 10, 111 Broue and Puig’s generalized characters, which are defined in [3], are relevant for the study of p-blocks with abelian defect groups. They will play an important role in this paper too. Let K be the algebraic number field containing the /Gi th roots of 1, let p be a prime ideal divisor of p in K, and let o be the ring of p-integers in K. Let B be a p-block and let (n, b) be a subsection associated with B. (n, b) is a pair such that ‘II is a p-element of G and b is a p-block of C(n) associated with B. In [3] (n, b) is called a (B, G)-Brauer element. Let x be an ordinary irreducible character in B. The function x”’ ‘) is the central function on G which vanishes outside the p-section of rc and which is such that x(X’ “(np) = x(npE,), where p is a p-regular element of C(n) and E, is the block idempotent of oC(n) corresponding to b. Following [I], for x, XI E B, we denote by m, xs (X3 ‘) the inner product of ~(“9 ‘) and x”“’ ‘) ~ rnpx?) is equal to the inner product of ~(~3 b, and 2’ by Brauer and &ma’s orthogonality relation. rnpxf) is called the contribution of (n, b) to the inner product (x, x’). By [l, (5B)],

Une correspondance de modules pour les blocs à groupes de défaut abéliens

Let b be a block of a finite group G with an abelian defect group P and an inertial quotient E. Let us denote by L the semi-direct product of P and E. If E is cyclic and acts freely on P−{1}, we prove that the stable categories of \(\mathcal{O}\)Gb- and \(\mathcal{O}\)L-modules are equivalent, as a consequence of a more general result, without any hypothesis on E, about partial covering exomorphisms relating \(\mathcal{O}\)Gb with a suitable twisted group algebra \(\mathcal{O}\)*\(\mathcal{O}\)

On p-blocks with abelian defect groups and inertial index 2 or 3, II

Let G be a finite group and B be a p-block of G with an abelian defect group and inertial index 2 or 3. In this paper we determine some generalized decomposition numbers of the block B.

The nilpotency index of the radicals of blocks with abelian defect groups

LetF be an algebraically closed field of characteristicp > 0, letG be a finitep-solvable group and letB be a block of the group algebraFG with abelian defect groupD. In this note we provide a precise formula for the nilpotency indext(B) ofJ(B), whereJ(B) is the Jacobson radical ofB. Namely, we prove that if $$D \cong Z_{p^{n_1 } } \times Z_{p^{n_2 } } \times \ldots \times Z_{p^{n_k } }$$ then $$t(B) = 1 - k + \sum\limits_{i = 1}^k {p^{n_i } .}$$

The Isomorphism Type of an Abelian Defect Group of a Block is Determined by its Modules

The Isomorphism Type of an Abelian Defect Group of a Block is Determined by its Modules

Note on a p-block of a finite group with abelian defect group

Let G be a finite group and B be a p-block of G with an abelian defect group D and with a root b in CD(D). Let Dl=CD(T(b)) and S1=lf^ ι\ where T(V) is the inertial group of b in NG(D). In [9, Theorem 1] we showed that the number k(B) (resp. l(B)) of ordinary (resp. modular) irreducible characters in B is equal to that of ordinary (resp. modular) irreducible characters in Sλ. In this paper we continue our study to show further properties on the p-block B^ Let K be the algebraic closure of the />-adic number field, o the ring of local integres in K and F be the residue class field of o. For a subgroup H of G, we denote by Tr# the relative trace map from (FG) to the center Z(FG) of the group ring FG, where (FG)={x^FG\hx=xh for any h^H}. For a ^-subgroup Q of G, we denote by SQ the Brauer homomorphism from (FG) Q to FCG(Q). The following is the main result.

Note on a p-block of a finite group with abelian defect group

Let G be a finite group and B be a p-block of G with an abelian defect group D and with a root b in CD(D). Let Dl=CD(T(b)) and S1=lf^ ι where T(V) is the inertial group of b in NG(D). In [9, Theorem 1] we showed that the number k(B) (resp. l(B)) of ordinary (resp. modular) irreducible characters in B is equal to that of ordinary (resp. modular) irreducible characters in Sλ. In this paper we continue our study to show further properties on the p-block B^ Let K be the algebraic closure of the />-adic number field, o the ring of local integres in K and F be the residue class field of o. For a subgroup H of G, we denote by Tr# the relative trace map from (FG) to the center Z(FG) of the group ring FG, where (FG)={x^FG\hx=xh for any h^H}. For a ^-subgroup Q of G, we denote by SQ the Brauer homomorphism from (FG) Q to FCG(Q). The following is the main result.

On p-blocks with abelian defect groups and inertial index 2 or 3, I

Let G be a finite group and B be a p-block of G with an abelian defect group and inertial index e. If e = 2, then the number of irreducible Brauer characters in B is 2. If p ≠ 2 and e = 3, then the number of irreducible Brauer characters in B is 3. The number of irreducible ordinary characters in B is also determined in both cases.