of the hardcover 1st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. To our loved ones 37 7. Group Cohomology and Change of Coefficients 44 3. PROJECTIVE RESOLUTIONS 47

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Abstract. Let k be an algebraically closed field of characteristic p > 0, and let G be a simple, simply connected algebraic group defined over Fp. Given r ≥ 1, set q = p r , and let G(Fq) be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology groupwhere L(λ) is the simple G-module of highest weight λ. Under certain very mild conditions on p and q, we are able to completely describe the first cohomology group when λ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when λ is a minimal nonzero dominant weight.