Group Theory 1. i. State the axioms for an equivalence relation ii. The relation n mod 3 divides the non-negative integers (i.e, n in Z such that n ≥ 0) into how many partitions? Show that n = 0 mod 3 is an equivalence relation. 2. Prove that, for any matrices, A, B and C: A+B=B+A And: A+(B+C)=(A+B)+C ( i.e., that the matrix addition is both commutative and associative) For simplicity, prove these properties using 2x2 matrices. 3. Prove that addition modulo n, written + is: i. Associative. ii. Commutative. There are two ways to prove these properties. Each way requires a definition or two: i. For n ≥ 2, 0 ≤ a, b ≤ n+1, a+ b= a+b if a+b< n a+n-n if a+b≥ n ii. Writing a for a mod n and (a+b) = a+ b, then: (p+ q) ≡ (p +q ) Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)? 4. Prove that addition modulo n, written + is: i. Associative ii Commutative. ( extra definations required : a for a mod n and (pà—q) = pà— q, so (pà— q) (p à—q ) 5. i. State the axioms defining a group - If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. - If (Z, à—) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative. - If (R, à—) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative. iii. In each case, deterimine whether the algebraic structure is a group. For each such group: - show how it satifies the group axioms - Draw the cayley table for the group and list the inverse elements i. For S=(0,2), a+2b ≡ (a+b) mod 2 and aà—2b ≡ (a à—b)mod2 a. (S,+ ) ( possibly an additive group) b. (S,∙ 2) (possibly a multiplicative group). ii. For S = (0,1,2) where n=2,3 and + and à— are defined as in the last part. a. (S,+n) ( possibly an additive group) b. (S,∙ n) (possibly a multiplicative group). Determine whether any of the groups is an abelian group. If any of them are abelian: i. state the conditions under which a group is abelian ii. show that the group is abelian 6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one 7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not. i.For any set, A, the set of one-to one and onto functions, f: A →A under composition ( written "◦"). ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under: a. Union b. Intersection iii. The set G=(a+b√5| a,b in Q) under addition and multiplication iv The set consisting of non-zero numbers under a. addition b. division v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table vi. The set (4,6) under multiplication modulo 12. draw cayley table vii. The set of real numbers under à—, where aà—b = 2(a+b) viii The set of real numbers under +, where a+b = a+b-10 ix. The set of rotational symmetries of a regular hexagon under composition x The following sets of permutations under composition i. (e,(12),(123),(1234)) ii. (e,(12), (34), (12), (34)) 8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e 9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order) 10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders 11.Find the subgroups of i. Z7 ii. Z8 iii Z9 12. i. determine which of the folowign are subgroups of under + a. (0) b. (-1,0,1) c. (n| n=10m for some integer m d.(p| p is a prime number e. (0,1,2,3,4) under addition modulo 5 ii. Determine which of the following are subgroups of under mulitiplication: a. (1, -1) b. (x |x=3, for some integer n c. (x |x=p/2ⁿ for some integers, p,n) d. (x| x=k 3 for some interger k

Group Theory Group Theory 1. i. State the axioms for an equivalence relation ii. The relation n mod 3 divides the non-negative integers (i.e, n in Z such that n ≥ 0) into how many partitions? Show that n = 0 mod 3 is an equivalence relation. 2. Prove that, for any matrices, A, B and C: A+B=B+A And: A+(B+C)=(A+B)+C ( i.e., that the matrix addition is both commutative and associative) For simplicity, prove these... click for more

Homomorphism

2.Let G be abelian of order n. If gcd(n;m) = 1, prove that f(g) = gm is an automorphism of G. (Note: Automorphism is just an isomorphism from G to itself.) 3. If f : Z7 ! Z5 is a homomorphism, prove that f(x) = 0 for all x 2 Z7. 4. Prove that in the group S10 every permutation of order 20 must be odd. 5. Suppose G is a group in which all subgroups are normal. Suppose further that x; y 2 G are such that gcd(jxj; jyj) = 1. (a) Prove that x¡1y¡1xy 2 hxi hyi. (b) Prove that xy = yx. 6. Let H be a subgrou of G, of ¯nite index [G : H] = k. Let C = fgH : g 2 Gg be the set of cosets of H in G. (a) Prove that for x 2 G the function ¸x(gH) = xgH is a permutation of C. (b) Let S(C) de... click for more

Modern Algebra - #1

Let G be the octive group in the following table and let H be the subgroup H = {e, B} of G. Find the district right cosets of H in G, and write out their elements. Please see attached for full question.

Modern Algebra - #2

Prove that mapping... is a homomorphism. Note: both groups are under addition. Please see attached for full question.

Modern Algebra - #3

Find the order of the element... Please see attached for full question.