“Study of symmetry”. Ok. But what’s the real intuition here? Why is symmetry so useful? Shortcut results? Work on one part and apply results to the others?

Symmetry has certain consequences, so you can see what is/isn’t possible given the initial conditions. (Can tile plane with squares, triangles, hexagons, but no other regular shape.)

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Remember there can only be 1 unique identity element per group.

Like a sudoku puzzle to figure out what the other elements must be

Subgroups can be found by shortcutting which properties we need to check. Can just check for closure, for example – since it inherits associativity, inverse, etc. from the parent.

For the rules of exponents, see $a$ as $e * a$. That is, we implicitly always start with the identity element and modify it from there. So $a^1 * a^{-1} = e$ is really saying “start with $e$, apply $a$, then undo $a$, and we’re left with $e$”.

Create a subgroup by picking an element (a) and just applying it to itself, for all items (in a finite set). That group will be closed. Keep applying the operation again and again.

Cyclic group – you can generate everything from just 1. (5 is also a generator from 1… well, it’s the INVERSE of 1 in a mod-6 group).

Idea… how do quaternions fit into groups? Similar to Klein?

Permutation groups… keeping track of WHAT you are doing [cycling, transposing] vs the final orientation. Similar to quaternions [the rotation, not the final orientation].

Cyclic group – whether you can “walk through” all the possibilities just by applying the first element.

Symmetry has many uses! One of the more fascinating ones is in theoretical physics. Because the universe has certain symmetries with respect to the laws of physics (e.g., the laws of physics are invariant under rotation of a reference frame), by necessity, there must exist a corresponding law of conservation (in this case, the conservation of angular momentum). Check out Noether’s theorem for more details. Finding all the transformations of a space which preserve a particular structure is one possible way to motivate group theory. A classic example is all the isometries of the plane which map the unit square to itself (this is the dihedral group of the square which has eight elements). Another possible way to motivate group theory is through permutations. (In particular, you can look at the permutations of roots of polynomials, which begins to get into Galois Theory.)

I’m not sure there is so much of an ‘aha!’ moment for group theory in the sense that when you learn it, it has already been codified for you. You don’t get to define what a group is, someone else just decided that already. This is actually a common trend in higher level math that eventually frustrated me a lot because I struggled to pick up the intuition behind why those axioms rather than some others. Maybe a nice way to see why the group axioms are reasonable is to play around with alternative axioms and see whether they ‘break’ in some sort of strange way (what if you drop associativity? what if you don’t include inverses? what if you demand commutativity? etc.)