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.
* e a1 a2 a3 a4
e e a1 a2 a3 a4
a1 a1
a2 a2
a3 a3
a4 a4

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 a1 a2 a3 a4
e e a1 a2 a3 a4
a1 a1 e
a2 a2 e
a3 a3 e
a4 a4 e

and then subsequently,

* e a1 a2 a3 a4
e e a1 a2 a3 a4
a1 a1 a2 a3 a4 e
a2 a2 a3 a4 e a1
a3 a3 a4 e a1 a2
a4 a4 e a1 a2 a3

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang