Instead, let’s break it down into simple terms and have a little fun along the way.
Well, in math-speak, a group is just a set of elements that follow certain rules (called “group laws”) when you combine them together. For example, if we take the numbers 1, 2, and 3 and make them into a group called G, then we can say:
G = {1, 2, 3}
Now let’s add some group laws to our little set of numbers. The first one is called “closure,” which means that if you combine any two elements in the group (let’s call them x and y), you should always get another element in the same group (let’s call it z). In other words:
x + y = z
The second law is called “associativity,” which means that when you combine three or more elements, it doesn’t matter how you do it as long as you follow the rules. For example:
(x + y) + z = x + (y + z)
Now computational group theory. This is where things get really interesting! Instead of just looking at groups on paper, we can use computers to help us solve problems and find new insights. One way to do this is by using algorithms which are basically a set of instructions that tell the computer what to do.
For example, let’s say you have a group called G with 10 elements (let’s call them g_1 through g_10). If you want to find out if two elements in this group are equal or not, you can use an algorithm like this:
for i = 1 to n-1 do
for j = i+1 to n do
if (g_i * g_j) == (g_j * g_i) then
print(“These elements are equal!”)
else
continue
endif
endfor
endfor
Now some real-world applications of computational group theory. One area where this is particularly useful is in cryptography which is the science of keeping secrets safe from prying eyes. By using groups to encrypt and decrypt messages, we can create secure communication systems that are virtually impossible to hack.
Another application of computational group theory is in physics specifically, in the study of quantum mechanics. By using groups to describe the behavior of subatomic particles, we can gain a deeper understanding of how the universe works at its most fundamental level.
Computational group theory: not as scary as it sounds (or looks on paper).