By David S. Dummit, Richard M. Foote

"Widely acclaimed algebra textual content. This e-book is designed to provide the reader perception into the facility and sweetness that accrues from a wealthy interaction among diversified parts of arithmetic. The booklet conscientiously develops the speculation of alternative algebraic buildings, starting from easy definitions to a couple in-depth effects, utilizing various examples and routines to help the reader's figuring out. during this method, readers achieve an appreciation for a way mathematical constructions and their interaction bring about robust effects and insights in a few diversified settings."

Covers primarily all undergraduate algebra. Searchable DJVU.

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Abstract Algebra (3rd Edition)

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2. 48 Representations and characters of groups 2. There is a correspondence between representations of G over F and FG-modules, as follows. (a) Suppose that r: G 3 GL (n, F) is a representation of G. Then F n is an FG-module, if we de®ne v g ˆ v( gr) (v P F n , g P G)X (b) If V is an FG-module, with basis B , then r: g 3 [ g]B is a representation of G over F. 3. If G is a subgroup of Sn , then the permutation FG-module has basis v1 , F F F , v n , and v i g ˆ v ig for all i with 1 < i < n, and all g in G.

Note that the regular FG-module has dimension equal to |G|. 6 Proposition The regular FG-module is faithful. Proof Suppose that g P G and v g ˆ v for all v P FG. Then 1 g ˆ 1, so g ˆ 1, and the result follows. 7 Example Let G ˆ C3 ˆ ka: a3 ˆ el. The elements of FG have the form Group algebras ë1 e ‡ ë2 a ‡ ë3 a2 57 (ë i P F)X We have (ë1 e ‡ ë2 a ‡ ë3 a2 )e ˆ ë1 e ‡ ë2 a ‡ ë3 a2 , (ë1 e ‡ ë2 a ‡ ë3 a2 )a ˆ ë3 e ‡ ë1 a ‡ ë2 a2 , (ë1 e ‡ ë2 a ‡ ë3 a2 )a2 ˆ ë2 e ‡ ë3 a ‡ ë1 a2 X By taking matrices relative to the basis e, a, a2 of FG, we obtain the regular representation of G: H I H I H I 1 0 0 0 1 0 0 0 1 e 3 d 0 1 0 e, a 3 d 0 0 1 e, a2 3 d 1 0 0 eX 0 0 1 1 0 0 0 1 0 FG acts on an FG-module You will remember that an FG-module is a vector space over F, together with a multiplication v g for v P V and g P G (and the multiplication satis®es various axioms).

If V is an FG-module, and W is a subspace of V which is itself an FG-module, then W is an FG-submodule of V. 2. The FG-module V is irreducible if it is non-zero and the only FGsubmodules are {0} and V. 52 Representations and characters of groups Exercises for Chapter 5 1. Let G ˆ C2 ˆ ka: a2 ˆ 1l, and let V ˆ F 2. For (á, â) P V, de®ne (á, â)1 ˆ (á, â) and (á, â)a ˆ (â, á). Verify that V is an FG-module and ®nd all the FG-submodules of V. 2. Let r and ó be equivalent representations of the group G over F.

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