Math 3310 - Algebraic Structures I 12/12/11 Final Ir Give precise denitions 4 of the following 7. (5 points each) 1 2 3 4 5 6 7 II. Work 3 of the following 5 problems. Determine if the statement is true or false. If true prove, if false r A group. i A subgroup. . An isomorphism. r An equivalence relation. r A vinyl r A eld. . An integral domain give a counter example. (10 points each) 1. 2. HL Work the following 3 of the following 5 problems. (10 points each) 1. The set C : {1 e 111:1 7 0) with the operation a: *9 : % is a group Let H and K be subgroups of Gr If H E K, then H is a subgroup of K. Let 01,02, and 03 be groups, and let f : GI H G; and 902 4v 03 he isomorphisms. The go f : G1 ~> G3 is an isomorphism. Let a be any element of nite order of a group G. If a has order 'n, and if a' : as, then 11 is a factor of r 7 3. i Let A be an integral domain. Let a 6 Ar If A has characteristic 1), and n . a : 0 where n is not a multiple of p, then a : 0. Let G be a group and let a,b,c E Gr Let 8 e G be the neutral element. Suppose that r; : cl. Then all : c if and only if abc : e. i Let A : {r E R : :c f 0 and 1 ii 1} Let G be the set of permutations G : {5,f1g} Where x) : 1/(1 71) and 9(1') : (:1: 7 1)/a:. Show that G is a subgroup of SA, and write the table of G. Let G be the group (10" : 71 E Z} with respect to multiplication. Prove that G E Z. (Remember that the operation of Z is addition.) . Prove that 'in Z,m ~ n if and only if im| : |n|,' is an equivalence relation on Z. Describe the partition associated with the equivalence relation. i Let A :Q and dene the operations a) and G) by aeb:a+b+1 aOb:ab+a+b_ Prove that A satises all the axioms to be a commutative ring with unity. IV. Prove 2 of the following 3 theorems. (10 points each) 1' Let (G, ) be a group and let S C G be nonempty. Suppose that the following hold. 1. S is closed with respect to multiplication. iii If a e S, then 111 6 S Then S is a subgroup of G. 2' Suppose an element a in a group has order n. Then at : e if and only if t is a multiple of n. 3' A ring has the cancellation property if and only if it has no divisors of zero