(d) A commutative ring R is a prime ideal of itself. (e) If p and q are primes, then there is a unique abelian group of order pq.
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Determine if True/False. Write True if it is always true. Otherwise, write False.
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- 15. Prove that if for all in the group , then is abelian.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.
- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.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.
- Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.9. Suppose that and are subgroups of the abelian group such that . Prove that .Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .14. Let be an abelian group of order where and are relatively prime. If and , prove that .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.