Just 6 short questions, and the questions are on the file ‘461’. I also uploaded the lectures for those questions. The time limit is due by this week Friday 11pm.

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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 2 Homework 2 1. For each equation in Zn find all solutions for x P Zn . (a) 3x ” 10 pmod 16q (b) 7x ” 9 pmod 18q (c) 4x ” 5 pmod 12q (d) 2x ” 6 pmod 12q 2. Find the orders of the following elements. (a) 9 pmod 51q in the group pZ51 , `q (b) 3 pmod 16q in the group pZˆ 16 , ˆq ? (c) 7 in the group pR,`

q ? (d) 7 in the group pRˆ , ˆq 3. Find the orders of the following elements in the general linear group GL2 pRq. ˆ ˙ ˆ ˙ ˆ ˙ 2 0 0 ´1 1 1 A“ , B“ , C“ 0 3 1 1 0 1 4. Let G be a group and a P G any element. (a) Show that if ak “ e then ordpaq divides k. (Hint: Write k “ ordpaqq ` r where 0 ď r ă ordpaq is the remainder.) (b) Suppose G is abelian, and b P G. Write m “ ordpaq, n “ ordpbq. Show that ordpabq divides the least common multiple of m, n. (c) Consider the group G “ te, r, b, g, o, yu from

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