Group theory studies the algebraic structures known as groups. So what exactly are these groups?

Imagine we have a paper square in front of us:

How many different ways can we move this square without changing how it looks? We can rotate it by any multiple of 90 degrees clockwise or counterclockwise or flip it across many lines. We can also just do nothing. To visualize what we’re doing, we can mark the corners of our square:

These are our symmetry transformations of the square. Notice that we can combine any two symmetry transformations to yield another symmetry transformation. Let’s visualize these compositions:

Here, R denotes rotations by the subscript degree, H is a flip over the horizontal line, V over the vertical line, MD over the diagonal line from top left to bottom right, and OD over the diagonal line from bottom left to top right.

We can begin to notice some properties:

- The composition of transformations is associative.
- The compositions of symmetry transformations yields a symmetry transformation.
- There is one element e such that A∘e=e∘A for every A.
- For every symmetry transformation A, there is a unique symmetry transformation A⁻¹ such that A∘A⁻¹=A⁻¹∘A=e.

These properties demonstrate that these symmetry tranformations form a group. This brings us to our definition of a group:

It turns out, groups are ubiquitous everywhere from physics to chemistry to even music. For example, Noether’s Theorem states that every conservation law in physics corresponds to a kind of symmetry–a group. The conservation of momentum corresponds to a translation in space, and the conservation of energy corresponds to a translation in time.