# 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.