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 wrote an article targeting the average lay person that teaches this way; https://rubberduckmaths.com/eulers_theorem
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.