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Xy = yx , for every x, y ∈ G , that is, its binary operation is commutative. If the group is abelian, it is usual to denote its binary operation with the + sign. We can understand the concept of group as a special case of groups with operators in ∅ (and as an action, the only one possible of ∅ on G ). −1 The element e will be called the left identity element or simply left identity of x and y will be called the left inverse of y . Analogously, we have a right identity element and a right inverse.

12 · 1 = 0} = {2, 4, 6, 8, 10, 0} = (2) . For q = 3 , m = 4 , we get {11·3 , 12·3 , 13·3 , 14·3 = 112 = 0} which, in additive notation is q = 4, For we obtain m = 3, {3 · 1, 6 · 1, 9 · 1, 12 · 1 = 0} = {3, 6, 9, 0} = (3) . 1·4 2·4 3·4 12 which, in additive notation is {1 , 1 , 1 = 1 = 0} {4 · 1, 8 · 1, 12 · 1 = 0} = {4, 8, 0} = (4) . For q = 6 , m = 2 , we get {11·6 , 12·6 = 112 = 0} which, in additive notation, is {6 · 1, 12 · 1 = 0} = {6, 0} = (6) . Finally, for q = 12 , m = 1 , we get {11·12 = 0} which, in additive notation, is {12 · 1 = 0} = {0} = (0) = O .

We distributed the integers in three boxes, where none of them were in two or more of the boxes, only in one of them. We labelled the boxes with three labels. We gave a group structure to the set of boxes by defining a binary operation. The reader verified that, indeed, we had a commutative group. We will call the boxes cosets and we will call the group a quotient group. In this case it is the quotient of Z “modulo” 3Z , which we will denote Z3 . Recall the concept of vector space quotient studied in Linear Algebra (see [Ll2]) and consider the additive part for the case in which G is an abelian group and H a subgroup of G , with x ∈ G ; we denoted as x + H the set {x + y|y ∈ H} .

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