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Q: 1. List all the subgroups of Z6 under addition mod 6 and draw the lattice diagram.
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Q: A cyclic group is abelian
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A: See the attachment.
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Q: 8. Prove that Zp has no nontrivial subgroups if p is prime. [#26, 4.5]
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Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.
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- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
- 6. Prove that if is a permutation on , then is a permutation on .Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].28. For each, define by for. a. Show that is an element of . b. Let .Prove that is a subgroup of under mapping composition. c. Prove that is abelian, even though is not.
- In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).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.Prove that if f is a permutation on A, then (f1)1=f.
- 19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.True or False Label each of the following statements as either true or false. Let A={ a,b,c }. The power set (A) is closed with respect to the binary operation of forming unions.[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]
