Write the Caley table of each of the subgroups of each abelian group of order 8.
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Write the Caley table of each of the subgroups of each abelian group of order 8.
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- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.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={ [ 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.Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined byExercises 35. Prove that any two groups of order are isomorphic.