Groups, Geometry and chalk dust

This is a diary of a founded research project in finite group theory - hopefully to be followed by a record of a journey through more beautiful maths.

[groupsgeometryandchalkdust]

The last few days that have passed marked an end to my summer research project (and an actual beginning to the long overdue summer holidays).

As it is often the case, thinking about things and coming up with ideas is the fun part of research – if you are a mathematician, what comes after is probably typesetting in LaTeX. Personally, being not used to that system (writing maths on paper and chalkboards is my preference) made it quite a journey to make all the tables with base sizes readable.

I am hoping to finish the report soon, currently doing the last remaining bits of computation. Afterwards, it will be time to read more about complexes made from pieces (simplices) of fixed curvature.

From now on, my posts will probably become less frequent and maybe a tad shorter. In my studies, I will be leaving finite groups to see a bit more geometry – looking forward to it :).

[groupsgeometryandchalkdust]

Note on Mathieu groups

As my project is ending, it is a high time to write something more about groups that kept me company through summer. First described in the second half of the 19th century by Émile L. Mathieu, and subsequently named after him, Mathieu groups are one of the families that comprises sporadic simple groups.

One can construct other groups in a similar vain to simple Mathieu groups – they can then also be called Mathieu groups and denoted by capital M with a suitable natural number in the lower index (simple ones have indexes 11, 12, 22, 23 or 24 – any other number means that we wandered off into a non-simple case).

Construction is where I find Mathieu groups to be particularly interesting – some arise as symmetries of Stainer systems or automorphisms of Golay codes, but both of those aren’t why and how Mathieu found them. His interests laid in multiple transitivity of actions of groups on sets, where an action is k-transitive if there is an element of our group that moves any k pairwise distinct points to any other k pairwise distinct points in the considered set.

Interest in k-transitive actions comes from their rarity. One of the most striking examples being the result of Cameron:  A finite group that is 5-transitive and neither symmetric nor alternating (so it doesn’t contain all possible permutations or all possible even permutations) is a Mathieu group on 12 or 24 points.

It turned out that in the project I used none of those constructions, instead viewing Mathieu groups as subgroups of respective symmetric groups – ATLAS provides permutation representations of each of them.

[groupsgeometryandchalkdust]

Actions and their equivalence

Group actions are, for me at least, an integral part of studying groups. I may have mentioned that intuitively they are natural (just as an adjective, not naturality in the sense of category theory) ways in which a group interacts with sets.

It turns out that a group can act on any set, if that action satisfies certain properties – no elements end up outside the set, identity does nothing (which should not be surprising) and acting by an element h and then element g, is the same as acting by their composition hg.

There are many interesting types of actions, and the ones I like thinking about the most are transitive. Transitivity means that, for any two points in the set on which the group acts, there is always an element of the group, that moves one to the other (the action allows us to ‘go anywhere’ in our set).

Being transitive is also important, because it allows a certain very useful result to hold – any such action of a group G is equivalent to some coset action. It follows, that although a group could transitively act on quite outlandish sets, that action is ‘the same’ (we can relabel points in our set to be cosets, and the action is compatible with that relabelling) as the group acting on cosets of one of its subgroups – which is often easier to describe and imagine.

As an aside, in the project I consider mostly actions on cosets, but I am becoming increasingly interested in how a group can act on shapes and topological spaces. Hopefully it will be something I explore in the coming academic year.

[groupsgeometryandchalkdust]

Computing in Magma

Since the beginning of my adventure with finding minimal base sizes, I have been using a computer algebra system. Reasons for that stem from, firstly, sporadic groups tending to be quite big (when it comes to the number of elements) and, secondly, many of them being given in the ATLAS by a permutation representation – which is easy to put into an algebra system but not so easy to work with on a sheet of paper.

Distributed by the Computational Algebra Group from the University of Sydney, Magma takes its name after an algebraic structure of the same name (which is similar to a group because it is a set with a closed operation, but no other conditions are imposed). This system is used in many areas of pure maths, like Number Theory, Group Theory, Representation Theory, and Linear Algebra, to name a few. 

When doing my project I use it to work with permutation groups – particularly, I have made friends with a command CosetAction(G, H). I also want to read about the particulars of the command IdentifyGroup(G), which allows identifying groups included in the Small Groups Library. Currently, that Library contains data on groups with up to 2000 elements (excluding 1024 – if your order is a large power of 2, you will get a lot of isomorphism classes) and some other groups of specific orders. If 2000 elements not seem like much, a fun fact is that there are more than 10 million non-isomorphic groups of order 512, so there are definitely many groups to be identified here.

I put the link to the website below, if anyone was interested - Magma is quite well documented (I found a 600-page handbook, only on commands concerning groups), so you can give it a try if you like.

http://magma.maths.usyd.edu.au/calc/

[groupsgeometryandchalkdust]

On groups and CFSG

As mentioned in the previous posts, groups can be thought of as structures describing symmetries. So, groups are just collections of transformations, that we can compose (for example first rotate an object and then reflect what we got along an axis of symmetry). And then it turns out our collection is just a set, and composition is an operation on its elements which satisfies some sensible and basic conditions – for example, we want to be able to do nothing or rewind whatever transformation we just did.

Sometimes it happens that there is a smaller group H, that lives inside another group G – taking a subset of our elements still gives us a well-behaved composition. Even better, a subgroup H might even be normal – and using it we may construct a quotient group G/H.

Simple groups are ones, that we can never make interesting quotients from - their normal subgroups are themselves or a set containing only identity (just a lonely “do nothing” element). Then quotient of a simple group by itself it’s just identity, and quotient by the identity is like leaving the group unchanged.

It often happens that the same group appears in different contexts, maybe getting differently labelled elements, or arising from a different construction. The notion of “sameness” we use in that case is being isomorphic – if two groups are isomorphic, that means we can rename their elements and the operation, and all that will result in us looking at exactly same structure. Because of that, when group theorists look at a group, they are looking at it up to isomorphism (as there is no point doing the same computation multiple times when we could just have changed labels).

As a result of thousand pages of proof, contributions of many mathematicians and half of a 20th century of work, all simple finite groups were classified (up to the “sameness” described above). The grand result that is the Classification of Finite Simple Groups (CFSG).

Sporadic groups are ones, that do not fit in into any infinite families of simple groups (groups in the same family are defined similarly, the only thing really changing is the coefficients in the construction). So, I guess some could call them a “classification bin” – but even if, I still find it incredibly interesting that there are only 26 of them (or 27 if one counts the Tits group). Doesn’t that number seem like coming out from nowhere?

(Sorry, I am still trying to find a balance of how much rigorous maths I should try to post – I am open to any comments or suggestions.)

[groupsgeometryandchalkdust]

First steps with the ATLAS

Before I started reading about group theory, I have seen many books and publications which all could be called atlases of some kind. Since, I have learned, that if you ask a group theorist about an atlas (in mathematical setting – I would still like to believe that mathematicians haven't forgotten about all the atlases one could find in their local book store), they will often think of the ATLAS.

The ATLAS of Finite Group Representations contains information on more than 700 groups and exists both as a physical publication, as well as in the form of websites. The online version, that I use because it’s easier to get your hands on it, has been prepared by Robert Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel Abbott.

For now, I seem to frequent only the pages on sporadic and alternating groups, but I am looking forward to exploring more. However, one thing I have noticed, is that notation of the ATLAS is not very well documented – rather it goes by the word of mouth. The tables on its pages are all full of dots, colons, and many different signs –  the meanings of which one works out often by asking older colleagues or mentors in hopes, they too have once asked and got the answer.

If you were curious about the ATLAS, I leave the link below. I also may create a glossary of notation because I still very much need it.

An Atlas of information (representations, presentations, standard generators, black box algorithms, maximal subgroups, conjugacy class repre

[groupsgeometryandchalkdust]

Welcome to this blog

Over the course of this summer, I will be working on a project founded by the University of Warwick undergraduate research scheme.

The research is based in Group Theory – an area of mathematics focusing on groups, which are structures describing symmetries. The groups I look at are the finite simple sporadic ones – most well-known being the so-called ‘Monster’ group.

My project begins with considering the Mathieu groups, the smallest groups of a kind described above. (They are not very small though – the one with the fewest number of elements is M11, having, 7920 of them.)

The aim of this project is to find minimal possible base sizes of actions of those groups – as actions are natural ways in which groups interact with their underlying sets and each other. Finally, base size is a notion used extensively when computing with groups.

The future posts will describe both the techniques and software that I am using, as well as contain some interesting facts I learned along the way.

Feel free to drop me a message or ask any questions.

With my hands already covered in chalk - Ala