Walter Feit

The computations of some Schur indices

AbstractLet χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letmp (χ) denote the Schur index of χ overQp, the completion ofQ atp. It is shown that ifx is ap′-element ofG such that

ON FINITE LINEAR GROUPS IN DIMENSION AT MOST 10

X_u \left( x \right) \in Q_p \left( X \right)

The K-admissibility of SL(2, 5)

for all irreducible charactersXu ofG thenmp (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(En), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz).

Examples of some Q-admissible groups

Let 0 be a finite group which has a faithful complex irreducible representation of degree d. Let p be a prime and let ‘

Finite linear groups and theorems of Minkowski and Schur

3 be a S,-subgroup of 0. If p > 2d + 1 then ‘p Q 0 (cf. [S]). Ifp > d + 1 then ‘p contains a subgroup !&, with 1 ‘p : ‘

Schur indices of characters of groups related to finite sporadic simple groups

J,, 1 d + 1 and y + 0. The main results of this paper provide a partial answer to this question. In particular, if 0 is simple the question is completely answered by these results. The following results are a slight generalization of those announced in [6].

The Q-admissibility of 2A6 and 2A7

Publisher Summary This chapter presents the assumption that G is a finite group with a faithful irreducible quasi-primitive unimodular complex representation of degree n. If n < 7, the structure of G is known by the work of Blichfeldt, Brauer, Lindsey, and Wales. This chapter presents some results that cover the cases that n = 8, 9, or 10. The work in case of n = 8 uses results of C. W. Huffman and Wales. The chapter presents the proof of a theorem as per which if G be a finite group with a faithful irreducible quasi-primitive unimodular complex representation of degree 10, either |G| = 2a·3b·5c·11d with d ≤ 1 or every composition factor of G is a known simple group or if 11|G| and |Z(G)|│2, then every composition factor of G is a known simple group.

TheK-admissibility of 2A 6 and 2A 7

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