The study of groups

Recently I watched a 3b1b video about group theory and how it relates to symmetry, like all knowledge I obtain through youtube my understanding is very shallow but I thought it was super interesting. In an effort to understand this topic better I'm writing my first blog post about it. One thing that Grant said that stood out to me the most was how symmetry seems to be the most fundamental concepts in math and physics and how group theory even relates to string theory. A symmetry group (not to be confused with symmetric group) for an object is the group of transformations for which an object is invariant, that is to say its properties remain unchanged, one example is rotating a square by any multiple of 90 degrees or reflecting it across its diagonals. Well, what's a group anyway?

a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.

The four group axioms

So the set and binary operation have to satisfy these axioms in order to be considered a group.

• Closure: a set is closed under an operation if performance of that operation on members of the set always produces a member of that set
• Associativity: for a binary operation, rearranging the parantheses will not change the result. $(2 + 3) + 4 = 2 + (3 + 4)$
• Identity: an element with respect to the binary operation which leaves any element of the set unchanged when combined with it. {Addition: 0, Multiplication: 1}
• Invertibility: an element that will 'undo' the effect of combination with another given element. {Addition: Negation, Multiplication: Reciprocation}

I think a good excercise is to see how these axioms are satisfied by a symmetry group and might give more insight into what exactly a symmetry group is.