- #1

Ulagatin

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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 ([tex]\text{for every }x, y \in G, x*y \in G[/tex]), associativity ([tex]\text{for every }x,y,z \in G, (x*y)*z = x*(y*z)[/tex]), identity ([tex]\text{there exists }e \in G \text{ such that }x \text{ }*\text{ } e = e \text{ }*\text{ } x = x \text{ for every }x \in G[/tex]) and inverse

([tex]\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[/tex]).

Furthermore, I understand that a group is denoted by [tex](G, *)[/tex]. I understand the concept of a subgroup (it is where a group H is "contained within a group G", mathematically expressed as [tex]H \subseteq G[/tex] where the group [tex](G, *)[/tex] has identity element e) and a proper subgroup [tex]\text{(where }H \neq {e}\text{ and }H \neq G)[/tex]. 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 [tex]\phi(n)[/tex] 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: [tex]\text{there are 8 subgroups of the group }Z/30^X\text{. Find them all. (List each subgroup only once!)}[/tex].

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