
# 2.2: Definition of a Group


Consider an object $$X$$ with some symmetries $$S$$. We've seen that we can compose any of the symmetries in $$S$$ and obtain another symmetry of $$X$$. We've also seen that these symmetries obey certain rules. We can now, at last, define a group.

Definition 2.1.0: Group

A group is a set $$S$$ with an operation $$\circ: S\times S\rightarrow S$$ satisfying the following properties:

1. Identity: There exists an element $$e\in S$$ such that for any $$f\in S$$ we have $$e\circ f = f\circ e = f$$.
2. Inverses: For any element $$f\in S$$ there exists $$g\in S$$ such that $$f\circ =e$$.
3. Associativity: For any $$f,g,h \in S$$, we have $$(f\circ g)\circ h = f\circ (g\circ h)$$.

An essential notion in mathematics is abstraction. Note that our definition certainly applies to any collection $$S$$ of symmetries of an object, but in fact there are other contexts where the definitions apply as well! The operation can be any way of combining two things in $$S$$ and getting another back; $$S$$ doesn't need to be a collection of functions, and the operation doesn't need to be composition. A group is defined purely by the rules that it follows! This is our first example of an algebraic structure; all the others that we meet will follow a similar template: A set with some operation(s) that follow some particular rules.

For example, consider the integers $$\mathbb{Z}$$ with the operation of addition. To check that the integers form a group, we need to check four things:

1. Addition takes two integers and gives another integer back. (Here we're checking the requirement that the operation is one from $$S\times S\rightarrow S$$. Notice that the the output of the operation is always in $$S$$! This is called closure of the operation.)
2. There's an identity element, $$0$$, where for any integer $$n$$, we have $$n+0=0+n=n$$.
3. Every integer $$n$$ has an inverse, $$-n$$, with $$n+(-n)=(-n)+n=0$$.
4. Addition of integers is associative.

Thus, the integers - with the operation of addition - form a group.

On the other hand, the set of integers with the operation of multiplication do not form a group. Multiplication does indeed take two integers and return another integer, and there is an identity $$1$$, and multiplication is associative. But not every element has an inverse that is also an integer. For example, the multiplicative inverse of $$2$$ is $$\frac{1}{2}$$, but this isn't an integer! Thus, integers with multiplication do not form a group.

An important note about inverses: An inverse means, roughly, that we can go back to where we started after applying an operation. Algebraically, this means we can cancel elements. When we have something like $$gh=gk$$, we can multiply both sides on the left by $$g^{-1}$$ to get $$h=k$$. We have to be careful to multiply on the same side on both sides, since groups aren't always commutative! If $$gh=kg$$, it doesn't necessarily tell us that $$h=k$$!

of $$X$$.)

Show that the symmetries of an equilateral triangle are not commutative. In other words, find two symmetries $$f, g$$ of the equilateral triangle such that $$fg\neq gf$$.