We consider a generalization of the Frobenius Problem where the object of interest is the greatest integer which has exactly j representations by a collection of positive relatively prime integers. We prove an analogue of a theorem of Brauer and Shockley and show how it can be used for computation.

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.

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A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught.