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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_forward23. 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 .arrow_forwardLet G be a group of finite order n. Prove that an=e for all a in G.arrow_forward
- let 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_forward14. Let be a homomorphism from the group to the group . Prove part a of Theorem : If denotes the identity in and denotes the identity in , then . Prove part b of Theorem : for all in .arrow_forward
- 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.arrow_forwardLet G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardFor each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.arrow_forwardLet G be the multiplicative group of units U20 consisting of all [a] in 20 that have multiplicative inverses. Find a normal subgroup H of G that has order 2 and construct a multiplication table for G/H.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,