Olaf Manz

The extended Golay codes considered as ideals

Abstract In this note we answer the following question in the affirmative: Is there a natural algebraic structure on the vector spaces containing the extended binary and ternary Golay codes such that the codes become ideals in these algebras? Our motivation was a note of J. Wolfmann, that describes the extended binary Golay code as the binary image of a principal ideal in a group algebra over the field with eight elements, and also a note of D. Y. Goldberg, that contains a related result for the extended ternary Golay code. In the following we construct those codes as ideals in the binary group algebra over the symmetric group H4 and in the ternary twisted group algebra over the alternating group u 4, respectively. Before we present our results, we are going to remind the reader of the definition of Golay codes as special QR-codes. Here, for obvious reasons, we restrict out attention to the binary and ternary case. We assume that the reader is familiar with basic notions in coding theory and in representation theory as well.

Brauer characters of q′-degree in p-solvable groups

Let G be a finite group and let p be a prime. We let Irr(G) and IBr,(G) denote the ordinary irreducible characters and the irreducible Brauer characters of G. Then p 1 x( 1) for all x E Irr(G) if and only if G has normal abelian Sylow-p-subgroup. For p-solvable G, this is a well-known result of Ito. Michler [9] recently proved the general case using the classification of simple groups. Also G has a normal Sylow-p-subgroup if and only if p j j?( 1) for all fl E IBr(G). This is due to Okuyama [ 1 l] for p = 2 and Michler [lo] for p odd. Suppose that G is p-solvable, q is a prime different from p, and q I/?( 1) for all j3 E IBrJG). We show that O”‘(G) is indeed solvable and that the q-length of G is at most two. In particular, since q-factors of G are necessarily abelian, it follows that the Sylow-q-subgroups of G are metabelian. For groups of odd order, these bounds were first obtained by Manz [S]. We also show that the p-length of P’(G) is bounded by at most

π-special characters

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Representations of solvable groups

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