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On Bounding the Schur Index of Induced Modules

On Bounding the Schur Index of Induced Modules ON BOUNDING THE SCHUR INDEX OF INDUCED MODULES B. BANIEQBAL 1. Introduction The purpose of the present paper is to prove the following result which was conjectured by B. Hartley. Recall that the index of a division algebra is the square root of its dimension over the centre. THEOREM. Let k be an algebraic number field and let H, G be finite groups such that H ^G. Let V be an irreducible kH-module such that V = kG ® V is an kH irreducible kG-module. Then the index o/End (K ) is bounded by a function of\H\. fcG If G is metabelian, a bound is 2\\{p— 1) where p runs over primes less than or equal to\H\. Note that the index in question is the Schur index of an absolutely irreducible constituent of V . Hartley arrived at the conjecture in the following manner. Let G be a locally finite group and H a finite subgroup. Let k be a number field and V be an irreducible fc./7-module. Under the assumption that V is an irreducible fcG-module, he shows that End (K ) is of finite dimension over its centre. This follows directly A:G from results in [3]. Bearing http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Bounding the Schur Index of Induced Modules

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/18.1.17
Publisher site
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Abstract

ON BOUNDING THE SCHUR INDEX OF INDUCED MODULES B. BANIEQBAL 1. Introduction The purpose of the present paper is to prove the following result which was conjectured by B. Hartley. Recall that the index of a division algebra is the square root of its dimension over the centre. THEOREM. Let k be an algebraic number field and let H, G be finite groups such that H ^G. Let V be an irreducible kH-module such that V = kG ® V is an kH irreducible kG-module. Then the index o/End (K ) is bounded by a function of\H\. fcG If G is metabelian, a bound is 2\\{p— 1) where p runs over primes less than or equal to\H\. Note that the index in question is the Schur index of an absolutely irreducible constituent of V . Hartley arrived at the conjecture in the following manner. Let G be a locally finite group and H a finite subgroup. Let k be a number field and V be an irreducible fc./7-module. Under the assumption that V is an irreducible fcG-module, he shows that End (K ) is of finite dimension over its centre. This follows directly A:G from results in [3]. Bearing

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1986

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