To kick things off: what is a group? Well, it’s not some sort of secret society or exclusive club that only math nerds can join. A group is simply a set of elements with certain properties. These properties include closure, associativity, identity, and inverses (which we’ll get to later).

Now closure. This means that if you take any two elements from the set and perform an operation on them, the result will also be part of the set. For example, if you have a group with the numbers 1, 2, and 3 as its elements, and you add those numbers together (using the standard addition operation), the resulting number will still be in the set. So 1 + 2 = 3 is an example of closure.

Next up: associativity. This property means that if you have three elements from the set and perform operations on them in a certain order, it doesn’t matter which way you do it the result will always be the same. For instance, let’s say we have the numbers 1, 2, and 3 again (because why not?). If we add 1 to 2 and then add that sum to 3, or if we first add 1 to 3 and then add 2 to the result, we will get the same answer either way. This is associativity in action!

Now identity. Every group has an element called the “identity” (or sometimes the “neutral” or “zero”). The identity is special because when you perform any operation on it with another element from the set, the result will always be that same other element. For example, if we have a group of numbers and 0 is our identity, then adding 5 to 0 gives us…you guessed it! 5.

Finally, inverses. Every element in a group has an “inverse” (or sometimes called the “opposite”). The inverse of an element is another element that, when combined with the original element using the operation defined for this set, gives us the identity. For instance, if we have a group of numbers and 5 is one of its elements, then the inverse of 5 would be -5 (because adding those two together gives us zero).

The basics of group theory in a nutshell. Of course, this is just scratching the surface there’s so much more to learn about groups and their properties. But for now, let’s all take a deep breath and appreciate how cool math can be (even if we don’t fully understand it).

Until next time, fellow mathematicians-in-training!