Let S = R\{-1} and define a binary operation on S by a*b = a + b + ab. Prove that (S, *) is an abelian group. [#7, 3.5]
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![2. Let S = R\{-1} and define a binary operation on S by a*b = a + b + ab. Prove that (S, *) is an
abelian group. [#7, 3.5]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F622f891a-3ac3-4b1a-95ba-531043fd4f39%2Fvwoemrn_processed.png&w=3840&q=75)
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- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.9. Suppose that and are subgroups of the abelian group such that . 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.Consider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.25. Prove or disprove that every group of order is abelian.
- 11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
- 14. Let be an abelian group of order where and are relatively prime. If and , prove that .Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)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.