1. Draw the cayley table for the group u(10) with respect to multiplication modulo and z8 with respect to addition modulo. What did you observe from the tables?
Answers
Answer:
Cayley Tables
When you studied multiplication in elementary school, you likely had to memorize multiplication tables. These tables had rows and columns of numbers as headings and products of those numbers in the interior of the table. Multiplication tables contain all the relationships between the numbers (at least as long as you only care about multiplication.)
A group is a set of elements closed under an associative operation that includes an identity and an inverse for each element. A Cayley Table describes the operation (i.e., the interaction between the elements) of a finite group. A Cayley Table is really the multiplication table for the group, except that the group operation may not necessarily be multiplication.
Example
Let's build the Cayley Table for the cyclic group of order 3 (that just means that it has three elements in the group). Although different symbols can by used, we will call the group elements 0, 1, and 2, and we will use the group operation of addition modulo 3. This means that we add the elements, but if we get a number higher than 2, we subtract 3 automatically. For example, 1+2=0 in this group since 1+2 gives 3, but then we subtract 3 to get 0.
Here is the Cayley Table for this group.
Cayley Table for Cyclic Group of Order 3
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To fill in the bottom row of the Cayley Table, added modulo 3 to get that 2+0=2, 2+1=0, and 2+2=1. Filling in a Cayley Table is easy as long as you know the group operation!
Properties of a Cayley Table
The Cayley Table gives all the information needed to understand the structure of a group. From the Cayley Table for the group above, we see what the elements are (0, 1, and 2). Did you notice that each row and each column contain each element? Like a correct Sudoku puzzle, this pattern is true for any correct Cayley Table. There are a couple of other things we can figure out about the group from its Cayley Table:
Because the 0 row is the same as the heading row and the 0 column is the same as the heading column, we know that 0 is the identity element of the group.
How do we find the inverse of a given element? Suppose we want to know the inverse of the element 1 in the cyclic group of order 3. Look in the 1 row and find the identity element 0. Then go up to find the heading of that column--this is the 2 column. Therefore, the inverse of 1 is 2.
Constructing a Cayley Table
To make a Cayley Table for a given finite group, begin by listing the group elements along the top row and along the left column. Traditionally, the identity element is listed first and the elements are listed in the same order left to right and top to bottom. This part is super easy!
Now all that remains is to fill in the interior of the table. To fill in the element in the row for element x and the column for element y, figure out what x*y is by thinking about the operation for that particular group. Do this for every pair of elements. Notice that x*y and y*x may not always be the same, so be careful about the rows and columns. (Here, the symbol * is representing the group operation, which may be addition, multiplication, composition, or something else depending on what the group is.)
Once you are done constructing a Cayley Table, how do you know if you have done it correctly or not? The only way to be truly sure is to double check each and every entry, but there are a few easy things to look for:
Every entry must be an element of the group. You can't 'make up' new elements as you go.
The row corresponding to the identity element (usually the top row) must match the heading row.
The column corresponding to the identity element (usually the left column) must match the heading column.
Each row must contain each group element exactly once. (Like Sudoku)
Each column must contain each group element exactly once. (Like Sudoku)
Example
You are asked to construct a Cayley Table for the group of symmetries of an equilateral triangle. A symmetry is a rigid motion that leaves the triangle looking the same as its initial position. It may help for you to cut out an equilateral triangle from a piece of paper to maneuver as you think about this. Label the corners of your triangle with the letters A, B, and C as shown and then label the corners on the back of the triangle so that they match the front
Step-by-step explanation:
Answer: A group is a collection of elements that are joined together by an associative operation that includes an inverse and identity for each element. A Cayley Table explains how a finite group functions (or, more specifically, how its elements interact). A Cayley table is actually the group's multiplication table, but the group operation need not always be multiplication.
Step-by-step explanation: Let's construct the Cayley Table for the order 3-cyclic group (that just means that it has three elements in the group). Although other symbols may be used, we'll refer to the group elements as 0, 1, and 2, and we'll employ the addition modulo 3 group operation. This implies that we sum the components, but if the result is greater than 2.
The Cayley Table for this group is shown below.
Cayley Table for the Third Cyclic Group
Modulo 3 was added to obtain the results 2+0=2, 1=0, and 2+2=1 in order to complete the bottom row of the Cayley Table. As long as you are familiar with the group operation, filling up a Cayley Table is simple!