Why is this Cayley diagram valid? I'm reading A Book of Abstract Algebra by Charles C. Pinter to give me an introduction to the topic, but one of the exercises has me confused. The following Cayley diagram is shown: 
But I can't understand why this isn't invalid - isn't each vertex connected to two vertices which represent the same element of of the group since the lines are all of the same type and undirected?
I'm sure the diagram is valid since it features on the front cover of the book!

The diagram you showed has bidirectional edges.  That implies that the generator they represent is its own inverse. 
But if the generator were its own inverse it couldn’t generate the 6 different states depicted on its own.
Therefore the 6-cycle you’ve shown needs unidirectional edges.

The key here is that edges in Cayley diagrams only represent generators and any generator of a cycle bigger than two must be unidirectional even though the group actions are all invertible.
(If we showed all the edges that connect nodes we’d have to connect each node [a “complete graph”, of your familiar with that vocabulary], which would be difficult to parse.)
