To kick things off: what is a group? Well, let me tell ya, it’s not some secret society or anything like that (although if you want to start one, go ahead). In math terms, a group is just a set of elements with certain properties specifically, they have an operation that can be performed on them and the result will always give you another element in the same set.

For example, let’s say we have the set {1,2} (which is called S for short). If we define our operation as “addition”, then we can perform operations like this:

S + S = {3}

So what does that mean? Well, when you add two elements from S together, you get a new element in the same set. In other words, 1+2=3 and 3 is also an element of our original set (which makes it a group).

But wait there’s more! A group has to have some special properties too. For example:

– Closure: If you perform the operation on any two elements in the set, the result will always be another element in that same set. In other words, if we add 1 and 2 together (which gives us 3), then adding 3 to anything else in our set should also give us an element of S.
– Associativity: This means that it doesn’t matter which order you perform the operations in they will always result in the same answer. For example, if we have three elements in our set (let’s call them a, b, and c), then performing (a + b) + c should give us the same result as a + (b + c).
– Identity: Every group has an element called “the identity” this is just another way of saying that there’s one number in our set that doesn’t change when we add it to anything else. In other words, if we have an operation like addition and we want to find the identity element (let’s call it 0), then adding any number from S to 0 should always give us back that same number.
– Inverse: Every group also has an “inverse” for each of its elements this is just another way of saying that there’s one number in our set that can be added or subtracted from a given element to get the identity (which we talked about earlier). For example, if we have an operation like addition and we want to find the inverse of 3 (let’s call it -3), then adding any number from S to 3 should always give us back that same number when we subtract -3.

That’s a basic introduction to group theory. If you want to learn more about this fascinating subject (or if you just need someone to talk math with), feel free to reach out to us anytime. We love nerding out over numbers and equations it’s our thing!