Hall and Higman's Theorem B is proved by constructing the representation in the group algebra. This proof is independent of the field characteristic, except in one case. Let R be an extra special r group. Suppose C_Aut(/?) is cyclic, irreducible faithful on R¡Z(R), and trivial on Z(R). The group CR and its representation theory have been widely studied and are of some importance. Let A be a field of characteristic q^r containing a splitting field for R. Then any faithful irreducible £[/?]-modulé V is absolutely irreducible and extends to CR. Hall and Higman [3] studied V\\c when C was a a group. Their result is proved using inequalities on the dimension of Hom^c^F, V). This proof also works if we only know q\\\\C\\. The central ideal of this proof also works when (a, | C\\)— 1 [5]. However, the count is quite different. There are character proofs of the result when (a, |C|)=1 [1]. Thompson gave a very pretty proof of Hall and Higman's original result using vertices and sources [4]. These proofs suffer from one or more of the following difficulties: (1) They are all tied to the characteristic a. (2) When q\\ \\C\\ they depend upon knowing all indecomposable k[C]modules. (3) They are indirect in that they do not construct a representation of CR. The theorem in all cases is that V\\c is a direct sum of copies of the regular C-module and one other module U isomorphic to k[C]/k[C]F where F= ZxeC x. The number of regular modules and the appearance of U are completely independent of the field characteristic. This makes (1) a strong objection. Objection (2) becomes important if we drop the condition that C be cyclic. Finally, a \"useful\" construction of a given module is often better than no construction at all. Received by the editors February 11, 1972. AMS (MOS) subject classifications (1970). Primary 20C15, 20C20; Secondary 16A64, 20H20.

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Hall-Higman type theorems. V

Remarks on isomorphic blocks

The equivalence of various generalizations of group rings and modules

Normal subrings and induced representations

Extending group modules in a relatively prime case

Isomorphisms of Clifford Extensions

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