MATH 113 Study Guide - Midterm Guide: Cyclic Group, Division Algorithm, Well-Order
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Write your answers on the exam, using both sides of the page if necessary. A group g is cyclic if there exists an element g g such that. G = {gi : i z}. (b) state the division algorithm for the integers. Let n be an integer and let d a positive integer. Then there exists a unique pair of integers (q, r) such that n = qd + r and 0 r < d. (c) use the division algorithm to prove that every subgroup of a cyclic group is cyclic. Suppose that h is a subgroup of g = {gi : i z}. If h = {e}, then certainly h is cyclic: h is {ei : i z}. Indeed, suppose that gi h. write i = qd + r as in the division algorithm. The group sn consists of the set of all bijections from the set.
