Let G be a finite abelian group and let a, bEG. Let m = ord(a) and b = ord (b). Let I = Icm(m, n). Then there exists some cEG such that ord (c) = l.
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- 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.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.