Cayley Tables

Given a finite group G with an operation on G, there is a way to visualize all the different elements of G and all the difrerent ways to combine them using the operation All the different elements of G and the different ways to combine them can be written in a single table, called a Cayley table, for easy reference. 

To create a Cayley table for a group G with operation, first list all the elements of G along the top and again down the left side of the table, being sure to list them in the same order in both places. In any list, the identity element e usually goes first. For two elements a and 6 in G, their combination a 6 is listed in the nth row and 6th column of the table, as shown in Table 5.1. 

Test yourself by creating the Cayley table for group Z5. The result should be the Cayley table presented in Table 5.4. 

The Cayley table for any finite group can be created. Just follow the same procedure and be sure to combine the group elements using the operation of the group. 

Example 5.2.1. Consider the group S with the composition operation o. Construct its Cayley table. 

To make it easier to list the group elements in the Cayley table, represent the elenients of S3 in the following manner ... 

Given these representations of the elenients in Table 5.5 presents the Cayley table for 53. 

Example 5.2.2 Create the Cayley table for the dihedral group the collection of symmetries of a regular 4-gon (a square), using the elements indicated below. 

To find the identity element of we must find the element eof H such that e a a 6 = a for each element a H. There are only four possibilities for e e must be a. S, 7, or 6. By using the Cayley table, it is readily seen that the only element that satisfies the stated condition is a. Thus the identity element e = a. 

Let s start with a itself. It is an element of H and must have an inverse such that a a = a. Locating a on the left-hand column of the Cayley table, find the element listed on the top of the table, such that a Oc. The only possible answer is that a = a. the element below the check-mark v/ in Table 5.8 on the left. A quick check shows that a a a also equals the identity a. 

Consider the collection of symmetries on a rectangle, as shown. You showed in Section 5.1 that this collection is a group. Create the Cayley table for this group. 

The work done in Sections 5.1 and 5.2 on groups and Cayley tables will now be used to investigate powers and orders of group elements. Once this is complete, the last piece will be in place so that the new check digit scheme promised at the beginning of this chapter can be developed. 

In determining the order of an element from a group, all that is needed is knowing how to combine elements from that group. The group operation could be presented by a Cayley table. Table 5.9 is the Cayley table for the group H = a,/ ,7,5 with operation. ... 

Consider this Cayley table for the group G = ,, , f,3 with the operation . 

In Exercise 5 from Section 5.2, a Cayley table was created for the group Dio (the collection of symmetries of a regular 5-gon). First each of the ten elements was denoted by a digit from 0 to 9 ... 

The resulting Cayley table for Dio, presented in Table 5.10 with operation provides a new way to combine the digits 0 through 9 that is nof commutative. 

In check digit calculations for the other schemes studied, digits were multiplied or added together For example, 3-4 = 12or3 + 4 = 7 would be calculated. In the Verhoeff scheme, digits are combined using the results of which are presented in the Cayley table (Table 5.10) for Dm- For example 3 4 = 2 and 8 3 = 5. 

German Bundesbank Check Digit Scheme In 1990. the German Bundesbank (Federal Bank) began using a scheme based on the Verhoeff check digit scheme [6]. While it does use the familiar Cayley table for Dio (Table 5.11). the Bundesbank s variation has two differences ... 

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