Morita Equivalent Blocks in Non-Normal Subgroups and p -Radical Blocks in Finite Groups

Abstract

ALet be a complete discrete valuation ring with unique maximal ideal J( ), let K be its quotient field of characteristic 0, and let k be its residue field \J( ) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the QGQ-th roots of unity, where QGQ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R ? o, kq. We consider the following question : when are A and B Morita equivalent ? Actually, we deal with ' naturally Morita equivalent blocks A and B ', which means that A is isomorphic to a full matrix algebra of B, as studied by B. Ku$ lshammer. However, Ku$ lshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Ku$ lshammer. Moreover, in the case H is normal in G, we get the same results as Ku$ lshammer ; however, he uses the results of E. C. Dade, and we do not. Introduction and notationLet (K, , k) be a p-modular system (see [16, p. 230]), that is, is a complete discrete valuation ring with unique maximal ideal J( ) such that J( ) l (π) for π ? , K is the quotient field of of characteristic 0, and k is the residue field \(π) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the QGQ-th roots of unity, where QGQ is the order of G. Then, in particular, K and k are both splitting fields for all subgroups of G (see [16, p. 230 ; 20, p. 13]). Let R ? ok, q. Now let H be an arbitrary subgroup of G, and let A and B respectively be blocks ( p-blocks) of RG and RH (here blocks of RG mean two-sided block ideals of the group algebra RG). We consider the following question, as in Ku$ lshammer [14] : when are A and B Morita equivalent ?As a matter of fact, we investigate naturally Morita equi alent blocks A and B as Ku$ lshammer did in [14]. Here, however, we assume only that H is just a subgroup of G, while he assumed in [14] that H is a normal subgroup of G.In §1 we consider p-radical blocks of finite groups where a result of Hida [9] is generalised (see [6, VI, §6; 21]). In § §2-4 we consider naturally Morita equivalent blocks in non-normal subgroups. In §3 we assume that H is a normal subgroup of G, and we give several results that are generalisations of Ku$ lshammer [14]. We also give a characterisation of an important subgroup G [B] of G which is due to Dade [4], though we do not use Dade's results [4]. In §4 we restrict ourselves to the case that A and B are naturally Morita equivalent of degree one, which is just the case that A and B are isomorphic blocks. Here also we do not assume that H is normal in G. We give an alternative proof of a theorem of Schmid [19].Throughout this paper, we use the following notation and terminology. Let k, , K and π be as above, and let R ? ok, q. Let A be an R-algebra. By an R-algebra A,

Clifford theory of characters in induced blocks

“…Then it holds n = 1. Therefore we get the assertion 40 by Theorem 4.1(7) of [HK99]. For the second case, a similar argument works by Proposition 2.4 of [HK99].…”

“…We know by Fong's result Theorem 5.5.16(ii) of [NT89] that b and B have a common defect group since p ∤ |G/N |. Hence, it follows from Theorem 7 of [Kül90] (see Theorem 3.5 of [HK99]) that B and b are naturally Morita equivalent of degree n for some integer n ≥ 1. Assume that the first case occurs, namely, that there is φ ∈ IBr(B) with φ N ∈ IBr(b).…”

Broué's Conjecture Holds for Principal 3-Blocks with Elementary Abelian Defect Group of Order 9

“…Since G is 3-nilpotent, G is 3-solvable of 3-length 1. So, it is well known that holds for G since B 0 k G and B 0 k H are isomorphic via restriction just as in (3.1) by a result of Isaacs-Smith [13] (see [11,Sect. 4]).…”

“…That is, X e ∼ = 2 . Hence, as we have done many times already, we get by [23, Lemma 2.1 (2)] that (14) Then, we finally know from (1)- (3), (7), (8), (11), and (14) that…”

“…Next, take I and B 4 . Let X be a B 3 -module appearing at the last column in (11). Let e be a primitive idempotent of B 3 corresponding to a simple B 3 -module 2 .…”