d_leet said:Is U(12) supposed to be the group of units modulo 12? If so then what elements are units in the group Z12? Maybe a better question is what is a unit?
A cayley table is a visual representation of a group's operation. It shows all possible combinations of elements within a group and the result of their operation. It is also known as a multiplication table or a combination table.
To construct a cayley table, you need to first identify the elements of the group. Then, you need to determine the operation for the group, which could be addition, multiplication, or any other binary operation. Finally, you fill out the table by performing the operation on each pair of elements. The result of the operation becomes the value in the corresponding cell of the table.
The main purpose of a cayley table is to provide a visual representation of a group's operation. It helps to understand the structure and properties of a group, such as the identity element, inverses, and closure. Cayley tables are also useful in determining whether a given set with a specific operation forms a group or not.
Yes, cayley tables can be constructed for any type of group, including finite and infinite groups. However, the size of the table may vary depending on the number of elements in the group. For large groups, it may not be feasible to construct a cayley table manually, and other methods, such as computer programming, may be used.
One limitation of using cayley tables is that they can only be constructed for binary operations, which means operations involving only two elements at a time. Additionally, cayley tables can be time-consuming to construct for large groups. They also do not provide any information about the structure of a group beyond the operation itself.