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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 6 Homework 5 1. Determine whether each map defined is a homomorphism or not. If it is a homomorphism, prove it, and also determine whether it is an isomorphism. (a) The maps φ1 , φ2 , φ3 ∶ pRˆ , ˆq Ñ GL2 pRq given by the following, for each a P Rˆ : ˆ ˙ ˆ ˙ ˆ ˙ 1 0 2a 0 1 a φ1 paq “ , φ2 paq “ , φ3 paq “ 0 a 0 a 0 1 (b) The map φ ∶ pZn , `q Ñ pCˆ , ˆq given by φpk pmod nqq “ e2πik{n . (Here i “ ? ´1.) (c) The map Cˆ Ñ GL2 pRq given by sending z “ a ` bi P Cˆ to: ˆ ˙ a b φpa ` biq “ ´b a 2. A while back on HW 1 you showed that the set Z equipped with the binary operation a ˚ b “ a ` b ` 1 defines a group. Show that this group pZ, ˚q is isomorphic to the usual group of integers with addition, pZ, `q. 3. Let φ ∶ G Ñ G1 be a group homomorphism. Define impφq “ ta1 P G1 ∶ a1 “

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