Continuing where Daniel left of with a discussions of groups of order 5.
A group with a prime order has certain qualities that are useful to know in figuring out its Cayley (multiplication) table. Most importantly, a prime group is always cyclic (can be generate by a single element and is abelian). This makes our job of filling out a Cayley table slightly easier. Let's start with our basic table, and fill out everything we know automatically.
- I have changed the notation slightly to emphasize that this is a group with a being the generating element with a^{5}=e.
* | e | a^{1} | a^{2} | a^{3} | a^{4} |
e | e | a^{1} | a^{2} | a^{3} | a^{4} |
a^{1} | a^{1} | ||||
a^{2} | a^{2} | ||||
a^{3} | a^{3} | ||||
a^{4} | a^{4} |
Now we know that elements can only appear once in a row and column and remembering that a^5=e, we can fill out the table as such:
* | e | a^{1} | a^{2} | a^{3} | a^{4} |
e | e | a^{1} | a^{2} | a^{3} | a^{4} |
a^{1} | a^{1} | e | |||
a^{2} | a^{2} | e | |||
a^{3} | a^{3} | e | |||
a^{4} | a^{4} | e |
and then subsequently,
* | e | a^{1} | a^{2} | a^{3} | a^{4} |
e | e | a^{1} | a^{2} | a^{3} | a^{4} |
a^{1} | a^{1} | a^{2} | a^{3} | a^{4} | e |
a^{2} | a^{2} | a^{3} | a^{4} | e | a^{1} |
a^{3} | a^{3} | a^{4} | e | a^{1} | a^{2} |
a^{4} | a^{4} | e | a^{1} | a^{2} | a^{3} |