> At this point, I didn’t have much else to say. Maybe because I started studying pure math later in life and I had to be more conscious about it, I have thoug

> At this point, I didn’t have much else to say. Maybe because I started studying pure math later in life and I had to be more conscious about it, I have thought a lot about what it means to understand and think well about mathematical concepts. I think the author misses the point of the question, "What is a group, actually?" after giving the formal definition of a group. The student was asking, "How am I to understand groups?" And more generally, understand mathematical concepts. Here are some tools I learned for understanding groups: - The informal definition. What's a group trying to model? - The formal definition. - Standard examples. What's the simplest? What are the "canonical examples"? Mathematicians love this question. See, for example (more links inside): http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures http://mathoverflow.net/questions/3242/canonical-examples-of... http://mathoverflow.net/questions/4994/fundamental-examples http://mathoverflow.net/questions/4994/fundamental-examples - What happens if we remove parts of the definition? Inverses? You get a monoid. Identity, too? A semigroup. What if you add additional properties? You get things like commutative groups, rings, etc. You'll want examples of these other structures, too. - The motivating history of groups. Who defined them, and why? - When mathematicians encounter a new object, the their first instinct is to classify it. Can we classify all groups? (No.) Are subclasses of groups that can be classified? Yes! Finite simple groups, finitely generated commutative and Lie groups are all hugely important examples. - Related to classification, what are the general properties of groups? What can you prove about them? This help develop an intuition for "how they behave". - The goal of math is to prove theorems. What are groups good for? What can you prove with them? This sets group theory in relation to other mathematical subfields. - As mathematical objects go, groups are relatively concrete, but for many objects, it can be a challenge to describe actual examples. How do you describe examples of groups? Tables, presentations (generators and relations). When do two presentations describe the same group? (Hard.) - What constructions exist on groups? Products, quotients, semi-direct (twisted) products, free products, fibered products, etc. Category theory suggests we should study not only groups, but relations between them. Groups form a category. What kind of category? What universal constructions hold in the category of groups? - What are the kinds of groups? Commutative, finite, finitely generated, Lie, matrix, simple, etc. All the above questions apply again! Canonical examples, history, informal definition, constructions, classifications, categories, etc.