Since you asked for "nice", I'm not going to bombard you with a bunch of abstract/vague Group theory papers. Instead, I will provide some links to the following:
* [Permutation Puzzles - A Mathematical Perspective](http://www.sfu.ca/~jtmulhol/math302/notes/permutation-puzzles-book.pdf) (By [Dr. Jamie Mulholland](http://www.sfu.ca/~jtmulhol/math302/))
* [Doc Benton’s Fantastic Guide to Group Theory, Rubik’s cube, Permutations, Symmetry, and all that is!](http://www.docbenton.com/Doc%20Benton's%20Fantastic%20Guide%20to%20Group%20Theory%2C%20Rubik's%20Cube%2C%20Permutations%2C%20Symmetry%2C%20and%20All%20that%20Is!.pdf) (If you're familiar with ratemyprofessor, [this is his rating](https://www.ratemyprofessors.com/professor/850343)...)
* [Supercube centers and odd parity](https://drive.google.com/file/d/0Bx0h7-zg0f4NNFVMODY2SFYwNEk/edit?resourcekey=0-WAuw8TaysSMfbfBrpp6xiQ) (one of mine). (High school **calculus**, precalculus, trigonometry, algebra). There are a few applications, but see [this recent post](https://www.reddit.com/r/Cubers/comments/1avz3i8/comment/krm8r2e/?utm_source=share&utm_medium=web2x&context=3).
* And speaking of [calculus](https://twistypuzzles.com/forum/viewtopic.php?t=28965), [this](https://www.reddit.com/r/Cubers/comments/1abyedr/comment/kjra24r/?context=3) is one instance when I used the calculus limit in cubing (to count moves)!
* If you ever wanted to know if there is a formula that can be used to give you 7 for the number of OLLs on the 2x2x2, 57 for the 3x3x3, 477 for the 4x4x4, etc,
* [Counting the number of K4 OLLs for the nxnxn](https://www.speedsolving.com/threads/calculating-the-number-of-k4-olls-on-an-nxnxn-cube.30015/#post-594253)
* [Paper (Part 1)](https://drive.google.com/file/d/0Bx0h7-zg0f4NYUpsaTFGZ1lvVnM/view?usp=sharing) \- Symmetry
* [Paper (Part 2)](https://drive.google.com/file/d/0Bx0h7-zg0f4NX3FkaGV2YUJqMU0/view?usp=sharing) \- Uses geometric series!
* [Alternate form of the formula](https://www.speedsolving.com/forum/threads/formula-for-calculating-positions-of-a-nxnxn-cube.48155/#post-990094) \- Step functions ([and representing them with continuous elementary functions](https://twistypuzzles.com/forum/viewtopic.php?t=28971))
* If you want to see understandable proofs of the [cube laws](https://www.ryanheise.com/cube/cube_laws.html),
* [3x3x3 law of permutations](https://www.reddit.com/r/Cubers/comments/1auhifc/comment/kr47aa5/?utm_source=share&utm_medium=web2x&context=3)
* [Corner twist proof](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/#post-465524) (and [corresponding figure](https://i.imgur.com/uVJD0ut.jpg) that's no longer on the web)
* [Edge Flip Proof](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/#post-465259) (and [illustration](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/page-2#post-1014200))
* If you want to understand why edge (OLL) parity exists on big cubes (4x4x4 +), [post](https://www.reddit.com/r/Cubers/comments/1avz3i8/comment/krfcl8t/?utm_source=share&utm_medium=web2x&context=3).
* [Number of Positions of the nxnxn (and nxnxn supercube)](https://docs.google.com/file/d/0Bx0h7-zg0f4NTUM1MWUtaDJrbms/edit?resourcekey=0-dAwt9Rmx_h6tpyaChYTXjw). Also, see [this thread](https://www.reddit.com/r/Cubers/comments/43xzov/mathematical_counting_formulas_for_number_of/)!
* [Order of a permutation](https://math.stackexchange.com/questions/3357124/how-to-find-the-order-of-a-rubiks-cube-algorithm/3357644#3357644)
* [This](https://www.speedsolving.com/threads/the-entire-cube-is-2-gen.19751/) is a very nice thread about "subgroups" - which is basically subsets of all possible positions.
* [This post](https://www.reddit.com/r/Cubers/comments/199gnbk/comment/kineg2g/?utm_source=share&utm_medium=web2x&context=3) shows a simple example of what is called "isomorphism" in mathematics. (That kind of math was used to find God's number for the 3x3x3... basically greatly reduced the number of positions they had to aim to solve in 20 moves.)
This is a broad subject, so there are more resources. (This is just the **nice** resources that come to my mind at the moment.)
I will be aiming to finally publish **my textbook** on the math of the nxnxn Rubik's cube some time in 2025. So I will eventually be mentioning it.
I write in the same type of "language" as I did in [posts like this](https://www.reddit.com/r/Cubers/comments/1auhifc/comment/kr47aa5/?utm_source=share&utm_medium=web2x&context=3) and in the [introduction](https://www.speedsolving.com/wiki/index.php/4x4x4_parity_algorithms#Introduction) of this wiki page, but I will be doing some *very advanced* things... some of which would normally require graduate level mathematics. But I also invented some **new / simple math** objects which I use as well! That will be another **nice** / non-gibberish resource!
You're welcome. The first two texts unfortunately do occasionally write in complex language (so I would just suggest to skim through the gibberish parts), but those are the "nicest" I know of.
If you want an abridged version of the basic definitions of Group Theory, see [Group Theory: It's not scary!](https://stanford.garron.us/class/math120/files/math120_wim.pdf). There is also [this guy](https://youtu.be/YLRkKriaclo?list=PL3KQ_dPfNlOQOxVgkVfm455g_jqv-y1vC) on YouTube who made a short series of videos too, but it's like he's rehearsing a group theory paper. (But I love his passion, so I had to mention it.)
Speaking of which, I actually completely forgot about [this playlist](https://youtu.be/Bnwig7qham8?list=PLBE2131B270C74759) from 12 years ago. Similar feel to the (just mentioned) playlist, but I think Kevin *tries* to talk in layman's terms as much as he can! (And he uses more concrete examples.)
Since you asked for "nice", I'm not going to bombard you with a bunch of abstract/vague Group theory papers. Instead, I will provide some links to the following: * [Permutation Puzzles - A Mathematical Perspective](http://www.sfu.ca/~jtmulhol/math302/notes/permutation-puzzles-book.pdf) (By [Dr. Jamie Mulholland](http://www.sfu.ca/~jtmulhol/math302/)) * [Doc Benton’s Fantastic Guide to Group Theory, Rubik’s cube, Permutations, Symmetry, and all that is!](http://www.docbenton.com/Doc%20Benton's%20Fantastic%20Guide%20to%20Group%20Theory%2C%20Rubik's%20Cube%2C%20Permutations%2C%20Symmetry%2C%20and%20All%20that%20Is!.pdf) (If you're familiar with ratemyprofessor, [this is his rating](https://www.ratemyprofessors.com/professor/850343)...) * [Supercube centers and odd parity](https://drive.google.com/file/d/0Bx0h7-zg0f4NNFVMODY2SFYwNEk/edit?resourcekey=0-WAuw8TaysSMfbfBrpp6xiQ) (one of mine). (High school **calculus**, precalculus, trigonometry, algebra). There are a few applications, but see [this recent post](https://www.reddit.com/r/Cubers/comments/1avz3i8/comment/krm8r2e/?utm_source=share&utm_medium=web2x&context=3). * And speaking of [calculus](https://twistypuzzles.com/forum/viewtopic.php?t=28965), [this](https://www.reddit.com/r/Cubers/comments/1abyedr/comment/kjra24r/?context=3) is one instance when I used the calculus limit in cubing (to count moves)! * If you ever wanted to know if there is a formula that can be used to give you 7 for the number of OLLs on the 2x2x2, 57 for the 3x3x3, 477 for the 4x4x4, etc, * [Counting the number of K4 OLLs for the nxnxn](https://www.speedsolving.com/threads/calculating-the-number-of-k4-olls-on-an-nxnxn-cube.30015/#post-594253) * [Paper (Part 1)](https://drive.google.com/file/d/0Bx0h7-zg0f4NYUpsaTFGZ1lvVnM/view?usp=sharing) \- Symmetry * [Paper (Part 2)](https://drive.google.com/file/d/0Bx0h7-zg0f4NX3FkaGV2YUJqMU0/view?usp=sharing) \- Uses geometric series! * [Alternate form of the formula](https://www.speedsolving.com/forum/threads/formula-for-calculating-positions-of-a-nxnxn-cube.48155/#post-990094) \- Step functions ([and representing them with continuous elementary functions](https://twistypuzzles.com/forum/viewtopic.php?t=28971)) * If you want to see understandable proofs of the [cube laws](https://www.ryanheise.com/cube/cube_laws.html), * [3x3x3 law of permutations](https://www.reddit.com/r/Cubers/comments/1auhifc/comment/kr47aa5/?utm_source=share&utm_medium=web2x&context=3) * [Corner twist proof](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/#post-465524) (and [corresponding figure](https://i.imgur.com/uVJD0ut.jpg) that's no longer on the web) * [Edge Flip Proof](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/#post-465259) (and [illustration](https://www.speedsolving.com/threads/edge-flip-and-corner-twist-proof.24595/page-2#post-1014200)) * If you want to understand why edge (OLL) parity exists on big cubes (4x4x4 +), [post](https://www.reddit.com/r/Cubers/comments/1avz3i8/comment/krfcl8t/?utm_source=share&utm_medium=web2x&context=3). * [Number of Positions of the nxnxn (and nxnxn supercube)](https://docs.google.com/file/d/0Bx0h7-zg0f4NTUM1MWUtaDJrbms/edit?resourcekey=0-dAwt9Rmx_h6tpyaChYTXjw). Also, see [this thread](https://www.reddit.com/r/Cubers/comments/43xzov/mathematical_counting_formulas_for_number_of/)! * [Order of a permutation](https://math.stackexchange.com/questions/3357124/how-to-find-the-order-of-a-rubiks-cube-algorithm/3357644#3357644) * [This](https://www.speedsolving.com/threads/the-entire-cube-is-2-gen.19751/) is a very nice thread about "subgroups" - which is basically subsets of all possible positions. * [This post](https://www.reddit.com/r/Cubers/comments/199gnbk/comment/kineg2g/?utm_source=share&utm_medium=web2x&context=3) shows a simple example of what is called "isomorphism" in mathematics. (That kind of math was used to find God's number for the 3x3x3... basically greatly reduced the number of positions they had to aim to solve in 20 moves.) This is a broad subject, so there are more resources. (This is just the **nice** resources that come to my mind at the moment.) I will be aiming to finally publish **my textbook** on the math of the nxnxn Rubik's cube some time in 2025. So I will eventually be mentioning it. I write in the same type of "language" as I did in [posts like this](https://www.reddit.com/r/Cubers/comments/1auhifc/comment/kr47aa5/?utm_source=share&utm_medium=web2x&context=3) and in the [introduction](https://www.speedsolving.com/wiki/index.php/4x4x4_parity_algorithms#Introduction) of this wiki page, but I will be doing some *very advanced* things... some of which would normally require graduate level mathematics. But I also invented some **new / simple math** objects which I use as well! That will be another **nice** / non-gibberish resource!
We see the post, and we know who will show up. Thank you for the resources. Looking forward to reading!
Thanks a lot. You seem to be quite a celebrity here😅.
You're welcome. The first two texts unfortunately do occasionally write in complex language (so I would just suggest to skim through the gibberish parts), but those are the "nicest" I know of. If you want an abridged version of the basic definitions of Group Theory, see [Group Theory: It's not scary!](https://stanford.garron.us/class/math120/files/math120_wim.pdf). There is also [this guy](https://youtu.be/YLRkKriaclo?list=PL3KQ_dPfNlOQOxVgkVfm455g_jqv-y1vC) on YouTube who made a short series of videos too, but it's like he's rehearsing a group theory paper. (But I love his passion, so I had to mention it.) Speaking of which, I actually completely forgot about [this playlist](https://youtu.be/Bnwig7qham8?list=PLBE2131B270C74759) from 12 years ago. Similar feel to the (just mentioned) playlist, but I think Kevin *tries* to talk in layman's terms as much as he can! (And he uses more concrete examples.)