Simple Groups of Infinite Matrices

By Peter Groenhout, University of Newcastle

While my research project is about ‘simple groups of infinite matrices’, it has been asked what a group is and why I would study them. Group theory can be interpreted as the study of symmetries, or transformations that preserve some sort of internal structure. These include the usual symmetries of geometric objects (such as rotational and reflective symmetries) and movement of infinite repeating objects – for example with an infinite 2D grid it can be moved up, down, left and right and still retain its gridhood.

One application of group theory is the classification of groups, which can be used to count distinct symmetries. For example, group theory shows that there are only 32 distinct 3D repeating patterns, which is important in the study of crystal structures. In the case of 2D repeating patterns (such as wallpaper patterns), it has been proven that there are only 17 distinct patterns.

This is an example of why I became interested in mathematics, namely that it is incredibly useful in explaining and making predictions about the natural world. I have since discovered that even the areas of mathematics that seem to be only of theoretical interest are still very useful – for instance complex arithmetic and finite fields are very important to electronics and modern cryptography.

With respect to my research topic, the ‘simple groups’ are a method of studying the internal structure of any group that can be decomposed into them. It has been shown fairly recently that finite simple groups can be classified into a small number of classes, and my research was intended to extend this to infinite simple groups even though it is unlikely they can be similarly classified.

My research project was successful in finding an infinite simple group and I am confident that this line of research will turn out to be both theoretically productive and have some real-world applications.

 

Peter Groenhout was a recipient of a 2018/19 AMSI Vacation Research Scholarship.

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