Group structure
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Group structure
Mathematical structure is the idea that different mathematical objects can share common features. A simple example of mathematical structure is the idea of congruency[Diagram goes here - download the original to see it.] Two triangles are congruent if it is possible to take one triangle, pick it up, and fit it exactly over the other. [Diagram goes here - download the original to see it.] What this diagram illustrates is that the study of structure - which is the study of what is similar between different objects - requires A description of the different objects; A description of the process by which one object is compared to another. In order to describe the two congruent triangles we could label their sides; congruency establishes a correspondence between these sides. [Diagram goes here - download the original to see it.] The correspondence is a mapping [Diagram goes here - download the original to see it.] There are an enormous number of different structures studied in mathematics. The idea of congruency is concerned with the similarities that exist between spatial structures. Algebra is generally concerned with the similarity of structures of objects that can be placed into sets. Sets are collections of objects. Numbers, for example can be collected into sets. However numbers are not the only mathematical objects that can be collected into sets. There are an extraordinary number of objects that may be collected. Geometrical processes, such as rotating and reflecting objects, can be collected into sets. Sets can also be finite or infinite in size. Set theory is the study of the similarity of structures that exist between objects merely because they are sets. Further mathematical structure arises when two elements of a set are combined in some way. Addition and multiplication of numbers are examples of operations on elements of a set . Because they take tow elements and combine them to form a third they are called binary operations. For example:- 2 + 3 = 5 is the binary operation (2, 3) ® 5 Similarly , 2 [symbol] 3 = 6 is the binary operation (2, 3) ® 6 In general, a binary operation is a function (mapping) from two elements to a third element. Thus new structures arise when we specify A set if elements, A binary operation defined no all elements of that set. However it will emerge that these requirement are not sufficient to define a structure meaningfully. Other conditions will need to be added. But firstly let us illustrate the idea of similarity of structure with the simplest structure of them all. Multiplication of positive and negative numbers. Set = The set made of the set of all positive numbers and the set of all negative numbers Operation = multiplication [Diagram goes here - download the original to see it.] Multiplication of +1 and -1. [Diagram goes here - download the original to see it.] Operation = multiplication [Diagram goes here - download the original to see it.] Consider a sheet of paper and two processes of (1) touching a sheet of paper -doing nothing to it; (2) turning the sheet over. We will label these two processes TOUCH and FLIP. The binary operation, O, will be the operation of doing one of these processes followed by the other. There are four possible combinations.TOUCH and TOUCH = TOUCH, TOUCH and FLIP = FLIP, FLIP and TOUCH = FLIP, FLIP and FLIP = TOUCH, The structure is represented by a table thus: Set = [symbol] Operation = Do one after the other [Diagram goes here - download the original to see it.] Intuitively, all the structures are identical as structures and differ only in the application of that structure to different sets. If we could discover what was common to a structure we would be then be able to use the results in any situation which possessed that structure. Mathematicians study structure because if they can solve a problem for a structure in general then the result will hold for all applications of that structure. Whilst our three structures: [Diagram goes here - download the original to see it.] Are intuitively different applications of the same structure, we have not proven this yet. But before we discuss what is required to prove the equivalence of such structures we must continue our discussion of the structures themselves. Recall that our study of structure begins with the specification for a given structure of A set of elements, G A binary operation defined on elements of that set, O The symbol (G,O) will specify such a structure. G stands for the set - Other symbols such as H,J,K could be used. O stands for the operation - Other symbols such as , ´, + could be used. Identity In order for the structure (G,O) to be a group the set G must possess an element e that is the identity under operation O. Consider, for example, multiplication of -1 and +1. The identity is the +1 since [Equation goes here - download the original to see it.] Operating with the identity leaves the element unchanged. Formally, Let G be a set and an operation on elements of that set. Then e is the identity element for that set under for all elements a in G. Thus, [Equation goes here - download the original to see it.] Note that for the set of all natural numbers under the operation of multiplcationthe identity is the number 1. For the set of all natural numbers under the operation of addition , the identity is the number 0 (zero). This illustrates that the identity element of a group depends upon the operation defined upon it. The same set can have different identities when different operations are defined upon it. (2)Inverses In order for the structure (G,O) to be a group for every element a in G there must be another element a-1 inG that is the inverse of a. If a-1 is the inverse of a, then a is the inverse of [symbol] .Inverses go in pairs. Two elements are inverses of eachother if, when combined together, the result is the identity element, [Equation goes here - download the original to see it.] or just [Equation goes here - download the original to see it.] Where e is the identity element. The identity element is always its own inverse. For the multiplication of +!, -1 the inverse of -1 is also itself, since-1 ´ -1 = +1 For the set of all integers , ¢, under addition the inverse of any number a is the integer -a , since [Equation goes here - download the original to see it.] and O is the identity for the set of all integers under addition. For example the inverse of 2 is -2 Note that the set of natural numbers, ¥ does not contain its own inverses. The set ¥ comprises all positive integers. ¥ = {0, 1, 2, 3,-----} so, for example, -2 is not in ¥. Thus ¥ is not in a group. In order to make ¥ into a group under addition we must extend it to the set ¢- the set of all positive and negative integers. But whilst ¢ is a group under addition, (as shall be fully shown), ¢ is not a group under multiplication. This is because, for example, the inverse of 2 under multiplication is 1/2, since [Equation goes here - download the original to see it.] and 1 is the identity under multiplication, and 1/2 is not an element of ¢. Hence in order to make ¢ into a group under multiplication we must extend it to the set Q = the set of all rational numbers. Then Q is a group under multiplication, as shall be shown. This illustrates that the same set, ¢, can be a group under one operation (addition) but not a group, under another (multiplication). (3)Closure In order for a structure [Equation goes here - download the original to see it.] to be a group, then for every pair of elements, a,b in G, the object that arises from the binary combination of these elements, , must also be an element of G. In other words, the binary operation does not lead us out of the set on which it is defined.The set G = {0,1} under addition is not closed since 1+1 = 2 and 2 is not in G. The set G = {1, -1} under multiplication is closed. To define a closed set under addition there are two approaches: (1) define an infinite set that does contain every result of adding two other elements of the set.(2) modify the operation of addition in some way so that finite sets of numbers are included. The first approach leads us once again to the structure (¢, +)- that is the set of all (positive and negative) numbers under the operation of addition. The second approach leads us to modular arithmetic. Students of mathematics should have already met the idea of modular arithmetic through the addition of angles. Since it is not possible to have an angle bigger than 360º(2 rad) whenever addition of two angles leads to an angle bigger than 360º(2 rad) we subtract 360º(2 rad) and start again. For example, 270º + 180º 450º 90º (mod 360º) The bracket indicates (mod 360º) indicates that the angle 0º is treated as equivalent to 360º and we are starting all over again. Modular arithmetic can be defined for any integer. The symbol ¢n stands for the group defined by addition module n. For example [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.] So the group table (the combination square) is [Equation goes here - download the original to see it.] Note that this is the same structure as [Equation goes here - download the original to see it.] illustrated earlier. For a second example [Equation goes here - download the original to see it.] With combination square: [Diagram goes here - download the original to see it.] These modular addition groups are also called cyclic groups. This is because in each case there is an element of the group that is combined with itself repeatedly (using the group operation (will generate every element in the group. For example in the case of ¢5 the group generator is the number 1. [Equation goes here - download the original to see it.] We denote the cyclic group generated by an element a of a group by . Thus ¢5 is an equivalent to the cyclic group under addition modulo 5 (4) Associativity In order for a structure [Equation goes here - download the original to see it.] to be a group, the operation, , must be associative. An operation is associative if, for any three elements a,b,c in G: [Equation goes here - download the original to see it.] Thus we ar enot obliged to bracket the elements when performing the group operation them and we can write unambiguously: [Equation goes here - download the original to see it.] Addition of integers is associative. For example. [Equation goes here - download the original to see it.] whatever way we add the numbers: [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] Thus the introduction of the additional bracket is not required and merely complicates the working. At this level failure of associativity is actually quite rare, and we are usually told to assume it. You can assume that addition and multiplication of numbers are associative. However one example of a structure which would be a group but for the failure of associativity is [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The group is closed; the identity is 0 and each element is its own inverse. But [Equation goes here - download the original to see it.] so associativity fails for this structure. Summary A group is an abstract mathematical structure comprising a set G and a binary operation o that satisfies the following properties (axioms). (1) Identity There is an element e Î G such that for any element a Î G [Equation goes here - download the original to see it.] (2) Inverses: For every element a Î G there is another element such that [Equation goes here - download the original to see it.] (3) Closure: For every pair of elements a,b, Î G there is a third element [Equation goes here - download the original to see it.] (4) Associativity For all elements a,b,c Î G [Equation goes here - download the original to see it.] Questions Questions on the definition of group structure ask you To construct a combination table for a given set and binary operation, To prove that a given structure is a group by verifying the axioms (i.e. identity, inverses, closure and associativity), To prove that given structure is not a group by showing that one of the axioms fails by providing a counter example
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Contents of
Group structure
1 Group structure
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