(e) Check as to whether the group S3 is abelian or not [NB: You need not check that S3 is a group as this is given]
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Q1e
Kindly answer correctly. Please show the necessary steps.
![(e) Check as to whether the group S3 is abelian or not
[NB: You need not check that S3 is a group as this is given]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabdd1821-d7b6-4171-a67e-edf70418727b%2Ff5f152ec-4326-4d46-a349-642de8d92ba6%2Fmmtnf6a_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.If a is an element of order m in a group G and ak=e, prove that m divides k.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.
- 9. Suppose that and are subgroups of the abelian group such that . Prove that .6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.
- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.