In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubikâs cubeâwhich weâll simplify to a 2x2 square for nowâ or anything, so long as they follow 4 specific rules, or axioms.
Axiom 1: All group operations must be closed, or restricted, to only group elements. So in our square, for any operation you doâlike turn it one way or the otherâyouâll still wind up with an element of the group. Or for integers, if we add 3 and 2, that gives us 1â4 and 5 arenât members of the group, so we roll around back to 0, similar to how 2 hours past 11 is 1 oâclock.
Axiom 2: If we regroup the order of the elements in an operation, we get the same result. In other words, if we turn our square right two times, then right once, thatâs the same as once, then twice. Or for numbers, 1+(1+1) is the same as (1+1)+1.
Axiom 3: For every operation, thereâs an element of our ground called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is 0. Not very exciting.
Axiom 4: Â Every group element has an element called its inverse, also in the group. When the two are brought together using groupâs addition operation, they result in the identity element, 0. So they can be thought of as cancelling each other out. Here 3 and 1 are each otherâs inverses, while 2 and 0 are their own worst enemies.
So thatâs all well and good, but whatâs the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, letâs expand our square back into a full-fledged Rubikâs cube. This is still a group that satisfies all of our axioms, though now with considerably more elements, and more operationsâwe can turn each row and column of each face.
Each position is called a permutation, and the more elements a group has, the more possible permutations there are. A Rubikâs cube has more than 43 quintillion permutations, so trying to solve it randomly isnât going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, thatâs exactly what most solvers do, even using a group theory notation indicating turns.
From the TED-Ed Lesson Group theory 101: How to play a Rubikâs Cube like a piano - Michael Staff
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