If m and n are positive integers, and if m is a factor of n, show that Zn contains a subgroup of order m. Does Zn contain more than one subgroup of order m?
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If m and n are positive integers, and if m is a factor of n, show that Zn contains a subgroup of order m. Does Zn contain more than one subgroup of order m?
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- Find all subgroups of the quaternion group.Exercises 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.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
- 4. List all the elements of the subgroupin the group under addition, and state its order.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 .