Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism, of order m and n, respectively. Show that there are MN abelian groups of order mn.
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- Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
- 13. Assume that are subgroups of the abelian group . Prove that if and only if is generated bylet Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- 15. Prove that if for all in the group , then is abelian.23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .25. Prove or disprove that every group of order is abelian.
- 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.Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.9. Suppose that and are subgroups of the abelian group such that . Prove that .