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- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.24. Let be a group and its center. Prove or disprove that if is in, then and are in.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.
- Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.