Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have taken introductory courses on groups, rings, fields and vector spaces and am currently taking one on modules. A common theme among such subjects are the three isomorphism theorems (as in, those found here

The naming of these theorems as "The First Isomorphism Theorem", "The Second Isomorphism Theorem" and "The Third Isomorphism Theorem" implies that they are in some way fundamental, basic theorems.

My question is, and I hope it's not too vague, in what sense are they fundamental and/or basic? Is it possible to formulate other, similar theorems, and if so, why would/are they not be grouped into this group of isomorphism theorems? I'm just trying to get a grasp of the big picture here.

share|cite|improve this question
up vote 3 down vote accepted

These theorems have generic names because they are "folklore", meaning they have been used so widely they are difficult to attribute.

Of course they need to be distinguished from each other, or else people might not know what isomorphism theorem you are talking about.

They are fundamental because they are proved generally in universal algebra for most algebraic objects. They ubiquity of their use also makes them fundamental. I think most algebraists use these theorems as unconsciously as breathing when thinking about algebraic objects.

I guess my feelings can be summarized this way: generic+ubiquitous+useful$\implies$ fundamental.

share|cite|improve this answer
Thanks, I don't feel enlightened just yet, but I suppose that will come with time and experience, your answer though will make sure my mind keeps the right view as to why the theorems are named the way they are. – Monty Gill Oct 17 '12 at 2:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.