Discussion group for abstract algebra? I'd be interested

[Ulagatin]

Hi everyone,

I've just finished year 11 here in Australia and I've been reading some notes on abstract algebra just out of curiosity. I have had a little difficulty grasping the concepts, and I've read up on some linear algebra (up to the point of Euclidean n-space - haven't yet read about eigenvectors and so on) but this hasn't really helped in terms of the concepts.

I understand that a group is defined with the axioms of closure ($$\text{for every }x, y \in G, x*y \in G$$), associativity ($$\text{for every }x,y,z \in G, (x*y)*z = x*(y*z)$$), identity ($$\text{there exists }e \in G \text{ such that }x \text{ }*\text{ } e = e \text{ }*\text{ } x = x \text{ for every }x \in G$$) and inverse
($$\text{for every }x \in G\text{, there exists an element }x' \in G\text{ such that }x\text{ }*\text{ } x' = x'\text{ }*\text{ }x = e$$).

Furthermore, I understand that a group is denoted by $$(G, *)$$. I understand the concept of a subgroup (it is where a group H is "contained within a group G", mathematically expressed as $$H \subseteq G$$ where the group $$(G, *)$$ has identity element e) and a proper subgroup $$\text{(where }H \neq {e}\text{ and }H \neq G)$$. What I do not understand at all is cyclic groups and the concept of order. Neither do I understand how to derive group tables, except that I guess it is based upon modular arithmetic. I think cosets make sense to me, but I would like to learn about the proof of Lagrange's theorem.

I believe the Euler phi function $$\phi(n)$$ from number theory is important in abstract algebra, but I do not see this relation.

I guess what I am asking for is some discussion on the basics of abstract algebra and a little collaboration - maybe a discussion group of sorts (pun intended). What I would like to be able to do is solve a problem such as: $$\text{there are 8 subgroups of the group }Z/30^X\text{. Find them all. (List each subgroup only once!)}$$.

I guess this group is the set of integers modulo 30, defined on multiplication, but I do not know how I would go about such a problem. If I can learn how to do this, I'll post a proof problem and perhaps a problem regarding order.

Hope we can have a productive and collaborative discussion on these basics of abstract algebra, because I'm quite interested in this, and I'm sure many others visiting the forum may be too.

Davin

 

[Ulagatin]

Oh, another question or two: what and how do cyclotomic polynomials work? I understand this has to do with fields. Also, polynomial rings look interesting - how would I go about solving a problem such as:

$$\text{Even though }x^6 + 3x^5 + 3x^4 +x^3 + 3x^2 + 3x + 4\text{ has no roots in }Z/5\text{, it has a repeated factor. Find it.}$$

Thanks again in advance - any help is greatly appreciated.

Merry Christmas to all those who celebrate it. Hope everyone has a prosperous and happy new year!

Davin

 

[VeeEight]

Multiplicative groups modulo n are a little different than additive groups because of the invertible elements. Here is a link: http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n.

A group is said to be cyclic if it can be generated by a single element (ie, G = {g, g², g³, ...}). It is called cyclic because you will get to a point where gⁿ=identity for some integer n, and then you start 'going in a circle'. Cyclic groups are interesting because they are all abelian and there are some so called 'fundamental theorems' that apply to them.

The order of a group is the size of it. The order of an element in a group is the smallest integer n such that gⁿ=identity. It is a well known group theory theorem that the order of an element divides the order of a group. The converse of this well known theorem is not necessarily true (that is, "if n divides the order of a G then G has an element of order n" is not a true statement). It is in fact true for prime numbers, however, and is called Cauchy's Theorem and is usually used to introduce the Sylow Theorems, which are very interesting group theory results.

Langrange's Theorem states that the order of a subgroup divides the order of the group it's contained in. The proof of Langrange's Theorem resides in the fact that all cosets of a subgroup have the same number of elements. Then you just use a little arithmetic to give the final consequence.

The connection between Euler's phi function and group theory is in Euler's Theorem and for counting the number of generators of a cyclic group of order n. There are of course more connections to study but these are usually the first two things discussed in a course in abstract algebra or discrete math.

 

[Ulagatin]

Thanks for the helpful reply. Will read the link in its entirety.

Cheers
Davin

 

[lapin]

The multiplicative group of Z_{30} has 8 elements (\phi(30). {1,7,11,13,17,19,23,29}. Here are the 8 subgroups {1,11}, {1,7,13, 19}, {1,17,19,23}, {1, 19}, {1,17,19,23}, {1,29}, {1}, and the whole thing.