Groups Continued: Katie - Rhea

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 a⁵=e.

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:

and then subsequently,