Introduction to Group Theory and Cyclic Groups

My goal in the next few posts is to talk about low order finite groups – that is, groups which contain a small number of elements.

My introduction to groups is going to be rather nonstandard. And it will be sketchy. Grab your favorite Introduction to Abstract Algebra or Introduction to Group Theory book. Suggestions:

• Fraleigh, “A First Course in Abstract Algebra”. A popular introductory text. I own two different editions.
• Dean, “Classical Abstract Algebra”, ISBN 0060416017. An excellent, if little known, introductory text.
• Dummit & Foote, “Abstract Algebra”. Written for undergraduates, with enough material for grad students. This was my main reading this time around.
• Schaum’s Outline of Group Theory. Cheap.
• Armstrong, “Groups and Symmetry”, ISBN 0387966757. An excellent undergraduate text with emphasis on the actions of groups on geometric figures. I’ve been using this book, too.

I think the cleanest starting point is to define a group as follows. We start with a set G, and a binary operation * on it. That is, given any two elements a, b of the set G, we have the product a*b, and that product is an element of G. Specifically, we require that the set be closed under the operation (or product). (That rules out, for example, the dot product of two vectors – because the dot product is not a vector, but a scalar.)