[btilly]

Not nobody. I know high school kids who would understand that. Furthermore I submit that someone who understands the phrases "linear algebra", "matrices", and "the trace of a matrix" is likely to understand that sentence. Anyone who knows what a "Lie group" is, DEFINITELY understands that sentence. (All of those phrases occur in the article without real explanation.)

As for what that sentence means, it means that given any two elements we can "multiply" them. Multiplication is associative, meaning that x * (y * z) is always the same as (x * y) * z. There is an identity simply means that an element we can call 1 exists such that 1 * x is always x. There is a multiplicative inverse means that for every x there is some element we can call x^(-1) such that x * x^(-1) = 1. By not commutative I mean that x * y doesn't have to be the same as y * x. (An example of non-commutative is that if you swap the first two items in a list then the first and third you get a different result than if you swap the first and third then swap the first two items.)

But still, you have answered my actual question.

What that site attempts to do is, at length, build some intuition about a topic, so that it can talk about it. By contrast my explanation just reminds someone who understands the fundamentals of the topic of the concepts so that we can start talking. So someone like you has a chance to understand something. Someone like me has to work to keep track of where they are, only to say, "You're just talking about group homeomorphisms."