Abelian groups
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The meaning of communtativity
The above properties of identity, inverses, closure and associativity are the minimum requirements to define an algebraic structure. A group is the starting point for the study of algebraic structure for this reason. However, there can exist algebraic structures with more structure, not less. One thing to consider adding to a group structure is the property of commutativity. Commutativity is the property that the order in which we do things does not matter. Some very familiar operations, such as addition and multiplication, have this property, and consequently, we take it for granted. For example the order in which we add two numbers does not matter - whatever order we do this we will get to the same total. Thus [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] so we can swap [Equation goes here - download the original to see it.] interchangeably. The same applies for multiplication [Equation goes here - download the original to see it.] Addition and multiplication are examples of binary operations and [Equation goes here - download the original to see it.] are groups. Let a and b be any two arbitrary real numbers. Then we have indicated that [Equation goes here - download the original to see it.] hold. Thus, both G and H are groups which the binary operation in question is commutative. They are commutative groups - we also call them Abelian groups. [Equation goes here - download the original to see it.] The property of commutativity is not trivial - by this, we mean, that there are structures that fail to be commutative. The usual standard example is matrix multiplication. For matrices it is not true that the order in which you multiply them together does not matter. In general [Equation goes here - download the original to see it.] We will show this with an example Equation goes here - download the original to see it.] Hence, adding the property of commutativity to a group turns it into a new structure meriting a separate name of Abelian group (which means, a commutative group). Example Prove [Equation goes here - download the original to see it.] is an Abelian group, where [Equation goes here - download the original to see it.] Solution Firstly, we remind you that the expression [Equation goes here - download the original to see it.] means the set of positive real numbers with the number 1 removed from it. To prove that this is a group we have to verify all the axioms - that is the four group axioms of identity, inverses, closure and associativity, together with the additional axiom of commutativity. Identity [Equation goes here - download the original to see it.] Therefore, the number e is the identity element Inverses [Equation goes here - download the original to see it.] Closure [Equation goes here - download the original to see it.] Associativity [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Commutativity [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence [Diagram goes here - download the original to see it.] is an Abelian group.
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Contents of
Abelian groups
1 Group structure
2 The meaning of communtativity
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