Project on Group Theory

I'm an IB student doing the Maths HL course, a Mathematical Exploration is mandatory for all students and should range between 6-12 pages.

Samples of these explorations can be found here and criteria on page 64 of this document.

I would like to do my exploration on Sets, Relations and Groups. This is out of the main syllabus and hence I have little to no knowledge of the material. I have a basic understanding of functions and relations, bijectivity, surjectivity and injectivity and an understanding of the group axioms, but beyond that. I have no knowledge what so ever. Are there any resources available that will provide me with the foundation of groups and relations and build up to some more advanced stuff?

Secondly, can you recommend any interesting applications of group theory and whether I would be better off discussing one application in exceeding depth for my exploration or cover two/three applications in deep, but not extremely deep in it?

Thank you! :smile:

Reply 1

tombayes

12

Original post by Zacken

I'm an IB student doing the Maths HL course, a Mathematical Exploration is mandatory for all students and should range between 6-12 pages.

Samples of these explorations can be found here and criteria on page 64 of this document.

I would like to do my exploration on Sets, Relations and Groups. This is out of the main syllabus and hence I have little to no knowledge of the material. I have a basic understanding of functions and relations, bijectivity, surjectivity and injectivity and an understanding of the group axioms, but beyond that. I have no knowledge what so ever. Are there any resources available that will provide me with the foundation of groups and relations and build up to some more advanced stuff?

Secondly, can you recommend any interesting applications of group theory and whether I would be better off discussing one application in exceeding depth for my exploration or cover two/three applications in deep, but not extremely deep in it?

Thank you! :smile:

First look up some basic types of groups: in particular: cyclic groups and symmetric groups. Also symmetry groups (different from symmetric groups!) are interesting e.g. dihedral groups.

Theory I would recommend:

Subgroups

Permutations/disjoint cycle notation/cycle shapes
even and odd permuations (some of this can be rather challenging)

Proof of Lagrange's Theorem (which is not in the hl maths syllabus but you might have done it class). In particular understand what a coset is.

Proof that the dihedral group of degree n has order 2n i.e. the number of symmetries of a regular n-gon is 2n. Also understand that the n reflections form a left(or right) coset to the rotation subgroup.

Proof of Fermat's Little Theorem using group theory

Group isomorphisms and isomorphism theorems

All this stuff is first year university maths so not totally out of reach. The above theory will give you the basic level of understanding required to actually apply group theory to other contexts.

(edited 9 years ago)

Reply 2

Zacken

OP

22

Original post by tombayes

x

Saved and noted. *drools* Those sound amazingingly interesting, positively delicious. :biggrin:

Unfortunately, I cannot seem to find material that starts from the basics online. Or at least in a structured manner. If you have any rescources or anything, I'd be very grateful. :smile:

Reply 3

tombayes

12

Original post by Zacken

Saved and noted. *drools* Those sound amazingingly interesting, positively delicious. :biggrin:

Unfortunately, I cannot seem to find material that starts from the basics online. Or at least in a structured manner. If you have any rescources or anything, I'd be very grateful. :smile: