When can you say that a group is abelian.
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When can you say that a group is abelian. Please express it in laymans term as much as possible
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- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.
- 15. Prove that if for all in the group , then is abelian.31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.Label each of the following statements as either true or false. Every cyclic group is abelian.