- #1
Ulagatin
- 70
- 0
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 ([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
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