The binary operation a ob=a is defined on Z. Select the correct statements. O The binary operation is closed. O The binary operation is abelian (commutative). O The binary operation has the inverse property. O The binary operation has the identity property. The binary operation is associative.
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- True or False Label each of the following statements as either true or false. An r-cycle is an even permutation if r is even and an odd permutation if r is odd.1. Let S = {a, b}. Define a binary operation ⋆ on S as follows:a⋆a=a a⋆b=a b⋆a=b b⋆b=b(d) Show that ⋆ has two distinct right identity elements, but no left identity element. So (S, ⋆) satisfies the third group axiom. (Note, the third group axiom does NOT demand that the structure have a unique right identity element.)(e) Show that (S, ⋆) does NOT satisfy the fourth group axiom by doing the following: Let a = eR be the right identity in Axiom 3, and then explain why some elements of S do not have a right inverse. Next let b = eR be the right identity element in Axiom 3, and then explain why some elements of S do not have a right inverse. At this point we know (S, ⋆) cannot be a group.Prove that if |G|= pq, where p and q are primes (not necessary distinct), then either G is abelian or Z(G) = {e}. (Hint: if G/Z(G) is cyclic, then G is abelian).
- Compute the index of 〈3〉 in ℤ12. Then write down all of the cosets of 〈3〉 in ℤ12. Finally, write a Cayley table for ℤ12/〈3〉. Is this quotient an abelian?Let a and b belong to some group. Suppose that |a| = m, |b| = n,and m and n are relatively prime. If ak = bk for some integer k,prove that mn divides k. Give an example to show that the conditionthat m and n are relatively prime is necessary.Suppose that A⊆G generates G. Prove that G is abelian if and only if any two elements of A commute with each other. (For the "⟸" direction, if you find you need to write a long induction proof as part of your solution, you are permitted instead to give a brief, informal description about how that inductive portion of the proof would go.)
- Prove that An is not abelian for all n > 4.EXERCISE 4.2.9. Let P be the set of people in a group, with |P| = p. Let C be a set of clubs formed by the people in this group, with |C| = c. Suppose that each club contains exactly g people, and each person is in exactly j clubs. Use two different ways to count the number of pairs (b, h) € P x C such that person b is in club h, and deduce a combinatorial identity.Generalize the argument given in Example 6 to obtain a theoremabout groups of order p2q, where p and q are distinct primes.
- show that the set Z ={0,+-1, +-2,...} with the operation of multiplication (a,b) map to ab is not a group Show that the operation (a,b) map to a*b =ab on the set of positive real numbers is not associative. for the permutation sigma belongs to S8 sigma=(3 7 4 2 6 8 1 5). Find the number of inversions I(sigma) sgn sigma =(-1)sigma , the decomposition into a product of independent cycles and the order for the permutation sigma belongs to S8 tau=(7 4 8 1 3 5 6 2). Find the number of inversions I(sigma) sgn sigma =(-1)sigma , the decomposition into a product of independent cycles and the order find the permutation U belongs to S8 such that sigma *U = tau where sigma=(3 7 4 2 6 8 1 5), tau=(7 4 8 1 3 5 6 2) are the permutation. decompose U into product of indepent cycles, find the sign and order consider the group of permutations Sn. Let A={a1,..., ak} subset of {1; : : : ; n}be a set of k less than equal to n distinct elements. Let SA subset of Sn be the set of permutationspreserving the…Let G = {1, 7, 17, 23, 49, 55, 65, 71} under multiplication modulo96. Express G as an external and an internal direct product of cyclicgroupsProve that if a, b ∈ G (group) and ab = ba, then |ab| is a divisor of the least common multiple of the numbers |a| and |b|.