So what is a group? Well, according to the textbook definition (which is always so much more exciting than anything else), a group is “a non-empty set G with a binary operation on G.” Wait, hold up did someone say “binary operation”? That sounds like something you’d find in a romance novel or maybe a really intense game of chess. But nope, it just means that there’s some way to combine elements from the group (which we’ll call “group elements”) and get another element back out again.

Now those three axioms that need to be satisfied for something to be considered a group. First up is associativity, which basically means that it doesn’t matter in what order you combine your group elements the result will always be the same. So if we have two group elements x and y, and another one called z, then (x * y) * z should give us the exact same thing as x * (y * z).

Next up is identity, which means that there’s some element in our group that doesn’t change anything when you combine it with any other group element. This is usually represented by an “e” or a “1”, depending on whether your professor likes to be fancy or not. So if we have x and e (where e is the identity), then x * e should give us back exactly what we started with namely, x itself!

Finally, there’s inverses. This means that for every group element x, there’s another one called “x-1” (or sometimes just “-x”) that will undo whatever x did So if we have x and its inverse (-x), then combining them should give us back the identity element e again!

And while that might not sound like much fun at first glance, trust me once you start playing around with groups and seeing all the cool things you can do with them, it’s pretty addictive!

But don’t just take our word for it. Check out some of these elementary consequences of group axioms (which are basically just fancy math terms for “cool stuff that happens when you combine groups in different ways”):

– If x and y are elements from the same group, then their product (x * y) is also an element from that same group! This might seem like common sense, but it’s actually a really important property of groups.

– The identity element e is unique there can only be one “e” in any given group. And if you combine the identity with anything else (like x), then you get back exactly what you started with!

– For every element x from our group, there’s another one called “-x” that undoes whatever x did This is known as an inverse, and it’s pretty important for all sorts of mathy reasons (like proving theorems we talked about earlier).

If you want to learn more about this fascinating subject, be sure to check out some of our other articles on elementary consequences of group axioms (which are basically just fancy math terms for “cool stuff that happens when you combine groups in different ways”). And if you have any questions or comments, feel free to reach out we’d love to hear from you!