A friend of mine is giving a lecture series on group theory and offered to teach me as well

A friend of mine is giving a lecture series on group theory and offered to teach me as well.

However I don't have any background in this kind of thing and am confused by the notation.

What does the "\{0}" mean for these last two invalid sets?

A \ B, when A are sets, means all elements of A that are not in B

So in this case, Z \ {0} is the integers excluding zero

Thanks, I get it.

So Z \{0} is a invalid group because many inverses are fractional.

But what does the subscript n mean on the last one?

Z_n denotes the set of integers mod n, i.e. the possible remainders when dividing by n.

Basically it's the set {0,1,...n-1}. But no zero in this case.

What is the inverse of the group Z_n \ {0}, * where n is prime?

3 isn't a complex number?
News to me.

Here's the problem when n is composite.

Suppose k is a factor of n. Then the equivalence ka = 1 (mod n) has no solutions, so k has no inverse.

e.g. could you solve the equivalence 2x = 1 (mod 8)?

You don't encounter this issue when n has no non-trivial factors.

oh whoops. i misunderstood what C is.

It should be invalid because the inverse of 0 is undefined, right?

man, i have so much to learn...

>It should be invalid because the inverse of 0 is undefined, right?
correct

Bruh. Just know your set definitions and group properties. Group theory is my absolute favourite topic but I only got to do it for 4 weeks a year ago. Can't wait to revisit it at uni.

moving on to Rings - is this an okay definition?

>A ring is a set with two operations attached. The set with each operation must form a group, and at least one of these groups must be commutative.

it can be proven that ax=1 (p) has a unique solution for each a. however you need to do some calculations to actually found out the specific inverses, you can't just write it down.

What do these square brackets mean?

Do you want the lecture notes?

The guy who wrote them is only 16, but hes one of them supersmart guys who's had a place on the IMO team since he was 9, unconditional offer at Cambridge age 15, etc etc

They seem pretty good to me so far.

to clarify, i have read the bit at the bottom but dont really understand it

the notation is already explained in that picture. Also for the Z[i] it means polynomials with integer coefficients evaluated at i.

>What do these square brackets mean?
It says it right on the picture...

Um, the picture you sent explained what the square brackets meant. Don't you understand it? Or am I misunderstanding your question?

Fuck mate Cambridge at 15? I'm doing pretty well at my age but shit like that makes me wonder if I could be as good as those guys if I applied myself as a young kid.

Yes, I dont understand what that means.

Is the set just {-1,0,+1}?

Who cares? Dont bother thinking about that stuff, it will only distract you.

I just try and learn as much as I can from him.

no wait, is the set Z[i],+,× actually {0,i,-1,-i,1}?

the set is any number of the form a+bx+cx^2+dx^3... (where a,b,c,d... are integers) evaluated at x=i, which in turn is equal to any complex number of the form a+bi (where a,b are integers).

I stated this somewhat unclearly. Any element in the set Z[i] can be expressed in the form a+bi (where a,b are integers). This is because i^2 = -1.

For example 1+2i+3i^2+4i^3+5i^4 = 1+2i + 3(-1) + 4(-i)+5(1) = (1-3+5) + i(2-4) = 3-2i.

Okay, I get that. Its kinda neat actually.

Okay, hope you have fun etc

What does the superscript on the N here mean?

I think it means natural numbers including zero, that is {0,1,2,3,...}. This is because sometimes N means {1,2,3,...} while other times it means {0,1,2,3,...}, there is no common agreement if to use 0 as a natural number or not. (This is just question of convention not really math).

Can you upload these notes somewhere? If they are images you can upload them as an album to imgur.

Thanks man! Would love to read them.

No. A ring will absolutely never form a group under multiplication, and you make no mention of distributivity.