If, like me, you're not a real mathematician but suffered through linear algebra and differential equations, you can still totally understand this stuff! I started off teaching myself differential geometry but ultimately had far more success with lie theory from a matrix groups perspective. I highly recommend:
My friends were all putnam nerds in college and I was not, and I assumed this math was all beyond me, but once you get the linear algebra down it's great!
My experience with groups and linear algebra is similar. I made real progress only after I got past the initial fear and intimidation, making a point of understanding those beautiful equations. Now I find myself agreeing with those who argue that mathematics education could profitably begin with sets and groups instead of numbers.
Super easy to explain sets and groups once you've learnt how modulus works too. Start with the additive group and how it behaves under mod m, then go into the multiplicative group and the differences it has and the show why x^y = 1 mod m for certain values due to behavior of the multiplicative group. It's reasonably easy to grok how those two groups work and this gives people an intuitive understanding for the additive and multiplicative groups and they can go further from there.
Hopefully it's helpful and gives people good intuition for this. Group theory is extremely fundamental and can and should be taught after basic arithmetic and modulus operations. There's really no reason it can't be taught in childhood.
>Super easy to explain sets and groups once you've learnt how modulus works too.
Wow you start going into the deep end and are already needlessly over-complicating everything.
I personally would have explained the concept of groups by writing the number symbols upside down and as words, count of things, etc. Then you force the students to prove the group properties. After that you should tell them to come up with a group isomorphism between the groups.
There is something off putting about being given definitions from a higher authority and having to wade through the mud and emerging with a poor intuition about the thing in question. Modular arithmetic is something that the students will have to learn on top of group theory, not something that acts as a learning aid.
It's kind of difficult to put into words, but the moment you manipulate any physical quantity, e.g. filling a kettle with water and emptying it, you are already deep into applications of group theory. The reason why it is possible to record physical quantities with numbers is that the physical thing you are measuring also obeys the properties of group theory.
What I'm trying to get at is that the definition of groups is that way, not just for a good reason, it must be that way, because otherwise it doesn't make sense.
There is an excellent series on youtube called "A friendly introduction to group theory"[1] which takes in my view a very intuitive approach of starting with symmetry groups. There's also "Group theory and the Rubik's Cube"[2] which teaches group theory starting with the symmetries of the Rubik's Cube. I personally think starting with symmetry groups and later on showing (via Cayley's theorem or whatever) that these are isomorphic to integers modulo n or general cyclic groups is the way to go to build intuition.
I am reading this with very little maths knowledge (since university 15 years ago) and I found this confusing:
"The multiplicative group of integers modulo n that we saw above gets more interesting when you consider a composite number such as 15 which has factors of 3 and 5. Repeated multiplication by 2 will never produce a multiple of 3 or 5 and this time there are only 8 numbers, {1,2,4,7,8,11,13,14} less than 15 that are not multiples of 3 or 5."
I understood the earlier example of "mod 3" because you only have {1,2} but then it becomes a lot more complicated but there's no explanation of it. Multiplying by 2 repeatedly under mod 15 only yields {1,2,4,8}.
After writing this, I saw you explained it a bit later in the document, so perhaps a note to that effect would help other readers.
I vividly remember first day of school after kindergarten in Spain. (3-4 years old?) Sets and Venn diagrams, how interesting and intuitive. Unfortunately it was arithmetic from then on.
Second this! And if you want a part memoir part history of this subject as it relates to physics (through Langlands Program) part ode to the beauty of maths, I recommend reading Edward Frenkel's Love & Math:
and if you went to school in maths but now have left that world, this book engenders an additional spark of nostalgia and fun due to reading about some of your professors and their (sometimes very difficult) journey in this world.
There is also Naive Lie Theory by Stillwell, which is targeted at an undergraduate level. I haven't read it yet, but it's been on my radar for a while.
it doesn’t say what a lie group is but it gets you down the road if understanding representations and what tou can do with them. dramatically easier than fulton and Harris for self-study.
https://www.amazon.com/Lie-Groups-Introduction-Graduate-Math...
and
https://bookstore.ams.org/text-13
My friends were all putnam nerds in college and I was not, and I assumed this math was all beyond me, but once you get the linear algebra down it's great!