There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z. Fill in the the multiplication table for the quotient group G/Z, and determine which of the two groups of order 4 is isomorphic to G/Z * || Z iz z iz jz kZ iZ jZ kZ
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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 .Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.15. 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 .
- Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .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 by
- 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 12.23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
- Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.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.14. Let be an abelian group of order where and are relatively prime. If and , prove that .