Referring to Example 13, verify the assertion that subtraction is not associativ

Question

Referring to Example 13, verify the assertion that subtraction is not associative. Show that [1,2,3] under multiplication modulo 4 is not a group but that [1,2,3,4] under multiplication modulo 5 is a group. Show that the group GL(2,R) of Example 9 is non-Abelian by exhibiting a pair of matrices A and B in GL(2,R) such that AB BA. Find the inverse of the element in GL(2,Zn). Give an example of group elements a and b with the property that a-1ba b. Translate each of the following multiplicative expressions into its additive counterpart. a2b3 a-2(b-1c)2 (ab2)-3c2 = e Show that the set [5,15,25,35] is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relation-ship between this group and U(8)? (From the GRE Practice Exam) Let p and q be distinct primes. Suppose that H is a proper subset of the integers and H is a group under addition that contains exactly three elements of the set [p, p q, pq, p2, q2]. Determine which of the following are the three elements in H. pq,p2,q2 p + q,pq,p2 p, p + q, pq p,p2,q2 p,pq,p2 In the notation of Example 16, verify that TabTcd = Ta+cb+d Prove that the set of all 2 times 2 matrices with entries from R and determinant +1 is a group under matrix multiplication. For any integer n>2, show that there are at least two elements in U(n) that satisfy x2 1. An abstract algebra teacher intended to give a typist a list of nine integers that form a group under multiplication modulo 91. Instead, one of the nine integers was inadvertently left out, so that the list appeared as 1,9,16,22,53,74,79,81. Which integer was left out? (This really happened!) Let G be a group with the following property: If a,b, and c belong to G and ab = ca, then b = c. Prove that G is Abelian (Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n = anbn. Is this also true for non-Abelian groups?

Explanation / Answer

the answer is: [9,9;10,8], the inverse of a matrix is easy, this is using the general linear group of invertible matrices.

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