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collection of equivalence classes forms a group This group is usually denoted by Z/6Z or Z6 or Z/6 It is a group having six elements; we call this a group of order 6 In general, the order of a group with nitely many elements is just the number of elements in the group Let H be the subset of Z6 consisting of E 0 , E 2 , E 4 Verify that this is a subgroup of order 3 Notice that 3 divides 6 The only other nontrivial subgroup is that consisting of E 0 and E 3 Notice that it has order 2 The number 6 has no nontrivial divisors besides 2 and 3 and the group Z/6Z has no other nontrivial subgroups Of course there is nothing special about the number 6 in the last example If k is any positive integer then we may declare that xRy if y x is evenly divisible by k The result is a group of order k denoted by Z/k Z or Zk or Z/k If the positive integer m evenly divides k (k = m p) then there will be one subgroup of order m and that group will consist of the elements E 0 , E p , E 2 p , , E (m 1) p When the context is understood we write the elements of Zk as 0, 1, 2, , k 1 We say that we are doing arithmetic modulo k or arithmetic mod k EXAMPLE 99 Consider the set Z4 Z4 This set may be conveniently thought of as the set of ordered pairs (x, y) where x, y Z4 De ne (x, y) + (x , y ) = (x + x , y + y ) where the addition is performed according to the group law of Z4 Then G = Z4 Z4 so equipped is a group of order 16 The number 4 divides the order of G, but now there is more than one subgroup having order (that is, number of elements) 4 One such subgroup is H = {(0, 0), (1, 0), (2, 0), (3, 0)} Another is K = {(0, 0), (0, 1), (0, 2), (0, 3)} Yet another is L = {(0, 0), (2, 0), (0, 2), (2, 2)} Now let G be a group of nite order Let g be a xed element of G Consider the set H of all powers of g: g 1 = g, g 2 = g g, g 3 = g g g, as well as g 1 , g 2 (g 1 )2 , g 3 (g 1 )3 , Of course g 0 = e It is easy to see that H is a subgroup of G We say that H is a cyclic group (subgroup) because it consists of powers of the single element g Let k be the order of H Then k will be the least positive integer such that g k = e Since H is a subgroup of G, we see that k = order H must evenly divide m = order G It follows (provide the details as an exercise) that g m = e This conclusion is so important that we display it in a theorem.

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