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- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.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.Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.True or False Label each of the following statements as either true or false. 7. If there exists an such that , where is an element of a group , then .