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ELEMENTS OF MODERN ALGEBRA
- 14. Let be an abelian group of order where and are relatively prime. If and , prove that .arrow_forwardLet G be a group of finite order n. Prove that an=e for all a in G.arrow_forwardLet R be a ring with unity and S be the set of all units in R. a. Prove or disprove that S is a subring of R. b. Prove or disprove that S is a group with respect to multiplication in R.arrow_forward
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.arrow_forwardlet Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.arrow_forwardLet G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.arrow_forward
- 9. Find all homomorphic images of the octic group.arrow_forwardSince this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardFor fixed integers a and b, let S={ ax+byxandy }. Prove that S is a subgroup of under addition.(A special form of this S is used in proving the existence of a greatest common divisor in Theorem 2.12.)arrow_forwardLabel each of the following statements as either true or false. Any idea of a ring R is a normal subgroup of the additive group R.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning