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.
- 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 .4. List all the elements of the subgroupin the group under addition, and state its order.