QUESTION 10 Use LaGrange's Theorem to prove that a group G of order 11 is cyclic.
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- 1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.9. Find all homomorphic images of the octic group.
- Exercises 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.Find all subgroups of the quaternion group.Exercises 21. Find all the distinct cyclic subgroups of the octic group in Exercise . 20. Construct a multiplication table for the octic group described in Example of this section.
- Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .True or false Label each of the following statements as either true or false, where is subgroup of a group. 4. The generator of a cyclic group is unique.
- True or false Label each of the following statements as either true or false, where is subgroup of a group. 5. Any subgroup of an abelian group is abelian.True or false Label each of the following statements as either true or false, where is subgroup of a group. 6. If a subgroup of a group is abelian, then must be abelian.31. a. Prove Theorem : The center of a group is an abelian subgroup of. b. Prove Theorem : Let be an element of a group .the centralizer of in is subgroup of.