Properties of groups
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Properties of groups
We have met the group axioms of (a) associativity, (b) identity, (c) inverses and (d closure. From these laws governing the definition and hence formal structure of a group various properties follow. The main properties are now described 1.Cancellation law Cancellation Law [Equation goes here - download the original to see it. [Equation goes here - download the original to see it.] Note that strictly we should write this using the symbol, o, or equivalent, for the operation in the group, thus In any group [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Throughout this section we will drop the use of the sign for the binary operation (for example, o) and we will drop other cumbersome notation that obscures the meaning.[Equation goes here - download the original to see it. Proof of the cancellation law [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] At each stage of the proof we state explicitly the property of the group structure that is being used. In other words, we justify each line. The proof of the second part of the theorem is almost identical to that of the first part. 2.There is just one identity element in any group. Proof [Equation goes here - download the original to see it.] 3.In any group each element has just one inverse Proof [Equation goes here - download the original to see it.] Likewise [Equation goes here - download the original to see it.] 4.In any group G, if a ÎG then (a-1)-1 = a Proof. [Equation goes here - download the original to see it.] What this says is that taking the inverse twice gets you back to where you started. 5 In any group[Equation goes here - download the original to see it. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] 6. For all elements a, b Î G there is another element x ÎG such that xa = b and an element y ÎG such that a = y Proof [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The second part of the proof follows similarly.From this theorem follows: 7. The Latin square property: the combination table for any finite group is such that each row and each column consists of a permutation of the group elements. Consider the row of the combination table corresponding to element a. The table shows the effect of combining a with every element of the group under the group operation. We have just shown that for any element b Î G there is another element x Î G such that b = a [Diagram goes here - download the original to see it.] This means that every element b Î G appears at least once in the row corresponding to the element a. But since there are exactly the same number of entries in a row as there are in a group set, it follows that every row contains all the elements of the set just once. That is, every row is just a permutation of the group elements.
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Contents of
Properties of groups
1 Properties of groups
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