Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm
 p (χ) denote the Schur index of χ overQ
 p
 , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that
 
$$X_u \left( x \right) \in Q_p \left( X \right)$$

 for all irreducible charactersX
 u ofG thenm
 p (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(E
 n), 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).

W. Feit, Characters of Finite Groups. VIII + 186 S. New York/Amsterdam 1967. W. A. Benjamin, Inc. Preis geb. $ 9.50 .

The computations of some Schur indices

/pdf/on-finite-groups-with-given-conjugate-types-ii-im8yjxapei.pdf

On finite groups with given conjugate types. II

The set of orders of elements in a finite group

Endliche zyklische Ebenen

An axiomatization of abelian hypergroups is given which allows for factor hypergroups to be developed. The hypergroup of characters and the hypergroup of conjugacy classes of a finite group are studied and compared. It is seen (using a theorem of Clifford) that if K is a normal subgroup of a finite group G, then the character hypergroup of G modulo that of G/K is isomorphic to the hypergroup of G-conjugacy classes of the irreducible characters of K.

/pdf/character-and-conjugacy-class-hypergroups-of-a-finite-group-27mdtdsidz.pdf

W. Feit, Characters of Finite Groups. VIII + 186 S. New York/Amsterdam 1967. W. A. Benjamin, Inc. Preis geb. $ 9.50 .

Citations

The computations of some Schur indices

On finite groups with given conjugate types. II

The set of orders of elements in a finite group

Endliche zyklische Ebenen

Character and conjugacy class hypergroups of a finite group

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Characters of finite groups

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