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4. Let G be a finite group and 4 : G F [3 resp element Z E FG to be 1 be a one dimensional representation. Define points Z (a) Prove that Z is an idempotent. G gEG [46-119. 2 141°3244 - X= - - (b) If R = FG, then show dimp Rx = 1. [3 poir 5. Suppose the character table of a finite group G has the following two rows [3 poi K1 K2 Ki3 K4 K5 Kif K7 X1 1 1 1 w² w w2 w 14121 X2 2 - 2 0 -1 - 1 1 1 where w = e2Tri/3 and the Ki denote conjugacy classes of G with K1 containing the ident (a) Determine the remaining entries of the character table. [4 po (b) Use part (a) to determine I G and the sizes of each conjugacy class. [2 pe (e(9,7) \ 2 =4

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Abstract Algebra Questions
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