Find the order of G. Find the order of i and -1.
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- Exercises 16. Assume that the nonzero complex numbers form a group with respect to multiplication. If and are real numbers and , the conjugate of the complex number is defined to be . With this notation, let be defined by for all in . Prove that is an automorphism of .9. Find all homomorphic images of the octic group.Exercises 9. Find an isomorphism from the multiplicative group of nonzero complex number to the multiplicative group and prove that . Sec. 15. Prove that each of the following subsets of is a subgroup of , the general linear group of order over . a.
- 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.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 2.1.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 .
- Let a and b be elements of a finite group G. Prove that a and a1 have the same order. Prove that a and bab1 have the same order. Prove that ab and ba have the same order.Prove that the group in Exercise is cyclic, with as a generator. Prove that for a fixed value of , the set of all th roots of forms a group with respect to multiplication.Find all homomorphic images of the quaternion group.
- 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)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 .