Morton E. Harris

Finite Groups With Product Fusion

In several recent papers concerning the classification of finite simple groups, the investigators were forced to treat subsidiary problems involving a group G whose Sylow 2-subgroup S is a direct product S = S, x S2 in which the fusion of 2-elements of G corresponds to that of the direct product of two groups having S, and S2, respectively, as their Sylow 2-subgroups. For example, in the study of groups, with Sylow 2-subgroups of type G2(q) or Psp(4, q), q odd, the determination of the structure of the centralizer of a central involution requires a solution of this subsidiary problem in the case in which S, and S2 are dihedral groups ([9]). The analogous problem arises with S, dihedral and S2 wreathed in Masons work on groups with Sylow 2-subgroups of type L4(q), q odd. Likewise in the analysis of groups with Sylow 2-subgroups of type Psp(6, q), q odd, two other such direct product problems arise in the course of determining the structure of the centralizers of involutions. It would therefore appear to be quite useful to have available an effective result about the structure of such a group G when the direct factors S, S2 of S are arbitrary. It is the object of this paper to establish such a general result. Suppose that we are trying to determine all simple groups G satisfying some set of conditions. In such a problem, it is very likely that one can reduce to the case in which the critical composition factors of the proper subgroups of G are of known type. Moreover, in those cases in which such a classification theorem has been obtained, specific properties of such composition factors have entered into the arguments. It is therefore reasonable and most likely necessary to impose some general conditions on appropriate composition factors of the proper subgroups of the group G under investigation. It turns out that a single assumption stated in terms of the notion of balance is all that is necessary. Moreover, we make this assumption only on the composition factors of certain subgroups of G which possess a Sylow 2-subgroup of the form T1 x T2with T, z Si, i = 1, 2.

Some results on coherent rings II

According to Bourbaki [1, pp. 62-63, Exercise 11], a left (resp. right) ^-module Mis said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3]. In this note, all rings contain an identity, all modules are unitary, and all ring homomorphisms preserve identities. If the underlying ring is non-commutative, all definitions and results will be given for the left side; the right side case will be immediate. The first results presented here concern a ring A with an ideal / which as a left ideal is finitely generated and an A/I -module M. They are used to derive necessary and sufficient coherence conditions on Aft and /for A to be left coherent. This theorem is used to show that the direct product of finitely many left coherent rings is left coherent and another application of this theorem is sketched. A result of [3] states that, if S is a multiplicative system in the commutative coherent ring A, then As must also be coherent. Here we show that, if every localization at a maximal ideal of a semi-local ring is coherent, then A is also coherent. Then an example of a commutative non-coherent ring is given whose localization at any maximal ideal is noetherian and hence coherent. Finally, some results on coherent modules over commutative rings are presented.

Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup

Let G be a finite group with a normal subgroup N that has a p-block b of defect 0. As is well-known, we may assume that b is G-stable. Then we show that there is an extension of G/N by a central cyclic p′-subgroup and a p-block of such that the blocks of G covering b biject with the blocks of that cover and corresponding blocks have several equivalent properties. These results extend the known results in the situation that N is a normal p′-subgroup.

Some Remarks on the Tensor Product of Algebras and Applications II

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Kulshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

A Remark on the Green Correspondence on G-Algebras

In the context of G-algebras, we prove that Green correspondent points satisfy some important properties that are suggested by the classical finite group Green correspondence.

Corrigendum to: “Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup”

At some point, after publication, the author realized that the proof of [3, Theorem 5.2] is incorrect. This proof incorrectly adapts the proof of [1, Theorem 4.8] since [3, (5.5)] is incorrect. Using the same proof outline, we correct the proof of [3, Theorem 5.2].

A Correction to “A Remark on the Green Correspondence on G-Algebras”

This note presents a counterexample to [1, Proposition 1(d)] that was communicated by Kulshammer. The proof of [1, Proposition 1(d)] is erroneous.

A characterization of odd order extensions of the finite simple Chevalley groups F4(q), q odd

Let p denote an odd prime integer and let y = pP where f is a positive integer. Let FJK) (=F,(y)) denote the finite simple Chevalley group of type (F4) over a field K of y elements. Then F,(K) has two conjugacy classes of involutions (cf., [S, Section 91). Let D denote an automorphism of K. Then 0 induces an automorphism of F,(K) (cf., [ll, Section lo]). In fact, , the cyclic subgroup of Aut(K) generated by 0, acts faithfully on F,(K) and one may form the natural semidirect product / F,(K). If 0 is an odd ordered automorphism of K, then G centralizes a Sylow 2-subgroup of F,(K) and F4(K) is an odd ordered extension of ii;(K) with trivial 2-core. In fact, any odd ordered extension of F,(K) with trivial 2-core is of this form (cf., 111, Section lo]), Let t be an involution in the center of a Sylow 2-subgr0u.p of F,(K) that is centralized by the odd ordered automorphism 0 of K. Then the centralizer C(t) of t in fcrFa(K) is a semidirect product ,U where %? denotes the centralizer of t in F,(K) and C(t) has trivial 2-core. Moreover, ?? contains a Sylow 2-subgroup of ,<:a) F%(K) and %? is isomorphic to Spin(9, K) (the universal Chevalley group associated with a root system of type (B,) over the field K). We shall prove the following result: