(4) Let G = {e, a, a², a³, aª, a5} be a cyclic group of order 6. (a) Construct the subgroup of G generated by the element a². (b) What are the possible orders of subgroups of the group G? (c) Find all subgroups of the group G.
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- Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .Find two groups of order 6 that are not isomorphic.Find all subgroups of the octic group D4.
- 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 In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?
- 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)3. Consider the group under addition. List all the elements of the subgroup, and state its order.Lagranges Theorem states that the order of a subgroup of a finite group must divide the order of the group. Prove or disprove its converse: if k divides the order of a finite group G, then there must exist a subgroup of G having order k.