I’m really excited to share my completed honours thesis! I’m honoured that my supervisors (and apparently the examiners too haha) were pleased with the result, and I’m humbled that my effort this year has led to graduating with first class honours — I owe it all to those who were there for me: my family, friends (especially my honours friends, church friends, Chris, and Wes), and God, whose love and presence sustained me through the busy times.
My research was in minimum bases for permutation groups. A permutation group is, loosely speaking, a collection of permutations of a set (essentially rearrangements of the set) which, when applied in succession, remains in the collection; the reverse permutation must also be in the collection, as well as the identity (do nothing) permutation. A base is a collection of points in the set being permuted, such that if a permutation in the collection fixes every point in the base, then that permutation must be the identity.
My goal was to talk about what is known about the smallest possible size of a base of a given permutation group, with use of the GAP computational language, and with original instructive examples interwoven throughout. I hope you find it interesting (or at least a pleasant read, which was definitely a focus and pride of mine) — and if you’re into group theory or maths in general, then I hope it may inspire you in your own work in some way. (You may want to watch the video below first, as it may clarify some basic content here in the thesis.)
Below is a recording of my final honours presentation. I motivated my research using the Rubik’s group (permutations of the Rubik’s cube), which is one of the many applications of permutation groups, and one that is close to my heart (thanks Wes). I tried to make it fun and engaging with the Rubik’s cube example, and a bit more understandable for a wider audience — I hope you will find it as interesting as I did!
While I might not be doing as much maths in the future, I hope to continue to review and share some more interesting things I’ve learnt along my mathematical journey. Despite going into full-time work not directly related to my field of study, my honours year was by no means a waste of time, and I am eternally grateful for this opportunity to spend 2022 studying something I’m passionate about. The following verse helped me get through these bittersweet feelings at the end of my honours year:
“There is a time for everything,
and a season for every activity under the heavens”
— Ecclesiastes 3:1 (NIV)
Thanks for coming along this journey with me, and I look forward to continuing in my new role at the Department of Education and Training and working towards better outcomes for education across our state!