# Group Theory Group Theory is all about codifying *[Symmetry](Symmetry.md)*. For instance, if we were to say that a face is symmetric, what we mean is that you can *reflect* it about line and it is left looking completely the same. It is a statement about an *action* that you can take. A snowflake is similar, but it has even more actions (rotations of various degrees, flipping along different axis, etc):  A collections of all of these actions taken together is called a **Group** (well, kind of):  We can see that the group of symmetries of a snowflake includes 12 distinct actions. It actually has a name, $D_6$. In general, there are a whole zoo of groups with no shortage of jargon to their names 😅), categorizing the many different ways that something can be symmetric.  #### Structure Preservation Now when we describe these actions there is always a certain **structure** being preserved. For example, there are 24 actions that we could apply to a cube that leave it the same. Those 24 actions taken together do indeed constitute a group. However, if we allow for reflections (in which case we don't care about preserving orientation) then we actually are dealing with a bigger group! If we were to let the faces of our cube move as well (in which case we are only viewing them as ever so loosely connected to begin with), we see that the number of actions we can take explodes!  The large size of this group reflects the *looser sense of structure* that each action preserves. #### What about no structure? The loosest sense of structure that we could possibly consider would be to look at any set of points and consider any way that we could shuffle them-any *permutation*-to be a symmetry of those points.  In the above case there are $6!$ ways that we could rearrange the points via permutation, leaving the structure unchanged (well there wasn't really any to start with). In contrast, consider if we imposed some structure via making these points the corner of a hexagon, where we now only consider the permutations that preserve how far apart each one is from the other.  ### Groups are actually more abstract Okay so far we haven't actually been working with Groups; rather, we have been working with **group actions**. Groups are a bit more abstract. What makes a group a group is how all of its different actions combine with each other. All possible ways that you can combine two elements of a group (i.e. two group actions) define a kind of *multiplication*; that is what really gives a group its structure. For instance, below, if we apply the action along the top row, followed by that along the left column, then the resulting action will be the same as what is in their shared square!  Now, if we replace each of these symmetric actions with something purely symbolic, well the multiplication table still captures the inner structure of the group, but now it is abstracted away from any specific object that it make act on, such as a square or roots of a polynomial.  This is entirely analogous to how the usual multiplication table is written symbolically, which abstracts away from the idea of literal counts.  Literal counts would arguably make it much clearer what is going on, but since grade school we have grown comfortable with the symbols.  These abstractions are less cumbersome, allow us to think about different numbers, and allow us to think about numbers in very different ways!  All of this is true of groups as well, which are best understood as abstractions above the group of symmetry actions! The key thing to keep in mind is the following relationship:  --- Date: 20210806 Links to: [Mathematics MOC](Mathematics%20MOC.md) Tags: References: * [Group Theory, abstraction - 3b1b](youtube.com/watch?v=mH0oCDa74tE)