let
Exercise16
For an integer
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Elements Of Modern Algebra
- Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.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_forwardLet G be a group of finite order n. Prove that an=e for all a in G.arrow_forward
- Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward24. Let be a group and its center. Prove or disprove that if is in, then and are in.arrow_forward
- 31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.arrow_forwardExercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,