Group Theory, Introduction
February 7, 2019 by Jayati Kaushik
The aim of this is not that you will see the many uses of group theory in daily life and instantly realize what an amazing subject it is. Nor is it to motivate it via it’s thrilling history (read about how Galois died).
The aim is to strip group theory of all this, and make you see what a wonderfully beautiful subject it is in its own right with no help from duels (seriously, read about Galois) or why it’s so important to physicists, or the fact that it was in nature and art before mathematics caught up to it (think symmetry).
After all, when does a mathematician study anything because it was “useful”? As Hardy once said, “I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Of course, given that a lot of his work was in number theory, he has since been proved wrong. The point is, mathematicians don’t study something because it will be “useful”, but because it is fun. We can’t help the fact that things we study are often useful. But let me assure you, that is rarely the primary goal; or even secondary… “Usefulness” it is probably last on the list. Riemannian geometry which started merely as a thought experiment; and yes, we do experiment; is exactly what Einstein needed centuries later for his theories.
Now, I am not saying the uses are not important, or that I will never go into the uses of this subject, but hopefully, I can show you that it is a beautiful, wonderful and fun subject even if one chooses to ignore it’s practical applications.
Right. To start with, we need a set. Any set. How about a set of ice cream scoops?
Let’s be slightly formal. Assume you own an ice cream shop; you lucky lucky hooman!. The set we want is set of all ice cream cups. Let’s say, that’s just how you store all your ice creams, in cups of the same size. To be sure: Each cup has one serving of one flavor. What can we do with these? Well, eat them. But hold on for a bit. What else can we do? What can one do with sets? We can union sets, compliment them, intersect them. What if we could combine two elements in a set? For example what does it mean to say chocolate ‘plus’ vanilla? To be sure, “plus” is not important. You can call it anything you like, for example dinosaur. What does chocolate ‘dinosaur’ vanilla mean? It could mean a any number of things. You being the shop owner are the judge of what it means(how does the power feel?). So, what can we do? Well, you can say, chocolate dinosaur vanilla just means one cup of vanilla and one cup of chocolate. But it could also mean anything that you choose it to be. For example, one cup of strawberry.
The idea now is to study these sets along with these operations.
I am gonna simplify things a bit. Ok, let’s do this.
You have the following flavors in infinite supply (your luck just keeps getting better):
Chocolate(C), Vanilla(V), Strawberry(S), and empty cups(E)(you never know when you might need one). Two flavors combined give one scoop of the third flavor.I write ‘+’ instead of dinosaur.
C+V = S
where x can be C, V, or S.
The way we have defined this operation? We take only two at a time and see how the combination works? Also notice, that the answer after we are done combining any two elements always stays within the set we defined.This is called a binary operation (Mathematicians are among the most creative people in world).
Another interesting thing to notice is that, E + anything gives us the same flavor of ice cream. E does noting to the element (As it should be, because an empty ice cream cup is absolutely useless! Keep them around though. Who knows?). Such an element is called the identity element.
OK, what if we wanted to combine more than two elements? We use brackets. For example,
V+S+C can be done in two ways: (V+S)+C = C+C=E, and V+(S+C) = V+V=E. But no matter how we choose to combine these, the result is the same: If you are too greedy, you get an empty cup.
Notice one more thing: No matter which ice cream we take; or which combination of ice creams we take; there is some flavor of ice cream we combine our result to get empty cups.
This my dear friends is what we call a group. A set and a way to combine elements of the set, such that all the above conditions are satisfied.
Now, go enjoy your ice cream. You deserve it.
Post written by Jayati Kaushik
Jayati Kaushik is a math student at the Technical University of Kaiserslautern, Germany with a passion for science communication. She is enthusiastic about learning and explaining mathematical ideas. She aims to show all people, young and old, how much fun mathematics can be. Ever ready to chat about mathematics, she can be reached at email@example.com.