Objectives of the poster project

• To explore independently a topic in group theory.
• To think about how to communicate your knowledge of this topic with other students in the class.
• To become proficient in LaTeX (if you haven't already).
• To think about approaches to project work in mathematics.

Potential topics and resources

1. The formulation of the group concept.
   The abstract definition of a group emerged from a long history of study of various examples of groups, and especially from detailed study of permutations and of solutions of polynomial equations in the 19th Century. An very good starting point for this topic (and many others) is the excellent MacTutor site on the History of Mathematics at the University of St Andrews. See this page in particular.
2. The frieze groups.
   A frieze is a broad horizontal band of decoration, often on the wall of a building and often with a periodic pattern. The frieze groups are basically the symmetry groups of such things, in an idealized mathematical context of course. Think of a single infinite strip of identical square or rectangular tiles. This has translational symmetries obtained by just sliding the entire strip along its length so that each tile lands in the position where another one started. There might be other symmetries as well, depending on what the individual tiles look like and whether they have rotational symmetries or refections. The frieze groups are basically the distinct groups that can arise this way, and there are only seven of them. This beautiful video from Atractor explains the seven groups and also highlights how group theory is connected to art and design. Another good reference for this topic is the wikipedia page on frieze groups.
3. The wallpaper groups.
   The wallpaper groups are the three-dimensional analogues of the frieze groups. They describe the possible symmetry configurations of repeating patterns on the plane (think of am infinite tiled floor). There are seventeen wallpaper groups. That there are not infinitely many is related to the fact that we can tile a floor with equilateral triangles or with squares or regular hexagons, but not with any other tiles that are shaped like regular polygons. This theme could be easily be the topic of multiple posters in the class - for example a poster could focus on just one of the seventeen wallpaper groups, or on a pair of them that are closely related, or on the ones that are based on a square or rectangular tiling (as opposed to a hexagonal one). Other possible themes include connections between group theory and art/design, or the appearance of these wallpaper patterns in archaeology and ancient history. If this general theme is of interest to several student groups, they could get together and think about how different groups could look at different aspects of it. Good references for this topic are the wikipedia page here, and these notes by Anneke Bart and Bryan Clair, from the EscherMath project at St Louis University.
4. The group of symmetries of the tetrahedron.
   The (geometric) groups of symmetries that we will discuss in our lectures are mostly the dihedral groups, which are the symmetry groups of regular polygons in 2-dimensional space. It is possible to apply the same ideas to three-dimensional objects, for example the platonic solids or regular polyhedra which are at least in some sense the "most symmetric" polyhedra. A tetrahedron is a triangular pyramid whose faces are all equilateral triangles. It has four vertices, four faces and six edges. The group of symmetries of the tetrahedron has 24 elements and it is isomorphic to the symmetric group of degree 4 (the group of all permutations of four objects). You could demonstrate this for example by showing that every permutation of the four vertices of a tetrahedron can be achieved either by a rotational symmetry or a reflection. Within this, the subgroup consisting of the rotational symmetries is also interesting. A More information here on Wikipedia.
5. The group of symmetries of the cube.
   As suggested for the tetrahedron above, you could think about presenting a description of the group of rotational symmetries of a cube, or the full group of symmetries. The cube has 24 rotational symmetries and they form a group that is a copy of S₄, the group of permutations of four objects. Identifying four objects related to a cube, and showing that the cube's rotational symmetries permute them in all possible ways is an interesting task and could be the basis for a very nice poster. The full symmetry group of the cube has 24 reflections as well as the 24 rotations. See wikipedia for some more detail.
6. The history of Lagrange's Theorem.
   A good reference for this topic is an article by Roth in the Mathematics Magazine, which can be found here in the JSTOR repository (let me know if you have any trouble accessing this, it should be ok if you are on a NUI Galway network). The version of this theorem that was presented by Lagrange in the 1770s long predates the modern definition of a group. It was concerned with changing a function by permuting its variables.
7. The number of generators of a cyclic group
   How many different elements of a cyclic group of order n are generators of the group? What does the answer have to do with the number n? How many generators does an infinite cyclic group have? See page 14 of the lecture notes and/or the wikipedia page on cyclic groups.
8. The converse of Lagrange's Theorem?
   Lagrange's Theorem says that if H is a subgroup of a finite group G, then the number of elements in H is a divisor of the number of elements in G. It is not true however that if k is a divisor of the order of G, then G must have a subgroup with k elements - i.e. the converse of Lagrange's Theorem does not hold in general. The smallest example for which it fails is A₄, the group of even permutations of 4 objects, which has 12 elements and has no subgroup with 6 elements. This group consists of all elements of S₄ that are either 3-cycles or products of two disjoint transpositions (and the identity element). A short and elementary proof of the fact that it has no subgroup of order 6 can be found here. This could form the basis for a poster on this theme. Alternatively you could demonstrate that the converse of Lagrange's Theorem does hold for some classes of groups, such as finite cyclic groups, finite abelian groups or finite dihedral groups. Or you could show that if p is a prime divisor of the order of a finite group G, then there is a subgroup of order p in G. This weaker version is called Cauchy's Theorem (not the only Cauchy's Theorem of course).
9. The exponent of a group.
   The exponent of a group is the least integer k with the property that x^k is the identity element for every element x of the group. If no such k exists then the exponent is infinite. A finite group always has a finite exponent. A poster on this topic could explain the concept of exponent, determine the exponent of the dihedral group of order 2n, explain that the exponent of a finite group is the least common multiple of the orders of the elements, and deduce that the exponent of a finite group is a divisor of the order. Note that the order of an element x in a finite group is the number of elements in the cyclic group generated by x; equivalently it is the least integer t for which x^t is the identity element.
10. The unit group of integers modulo n.
    For a positive integer n write _n for the set of congruence classes of the integers modulo n. There are n elements, represented by 0,1,...,n-1, with additon and multiplication modulo n. These elements do not form a group under multiplication (because the zero element does not have an inverse). For which values of n do the nonzero elements of _n form a group under multiplication? If this does not occur, the subset consisting of those elements which do have inverses will always form a group, called the group of units of integers modulo n. How many elements does this have? How are they identified? A poster on this could include the multiplication tables (for all of the non-zero classes or just for the invertible ones) for some examples such as n=4,5,6,7,8. Have a look at this wiki page for a nice account of this topic.
11. Infinitely generated groups
    Infinite groups can have finite generating sets (like the additive group of integers which is generated by a single element) or not (like the additive group of the rational numbers). A poster on this topic could give some examples, with explanation, of infinite groups that do not have finite generating sets.
12. Generating sets of symmetric groups
    The symmetric group of degree n is the group of all permutations of a set of n objects. It has order n!. A poster on this topic could explain why this group has generating sets with just two elements, or explain (with examples) why the symmetric group is generated by transpositions (elements that just swap two objects).
13. Even and odd permutations
    Every permutation of n objects can be written (in lots of ways) as a product of transpositions. A permutation that can be written as the product of an even number of transpositions cannot also be written as the product of an odd number of transpositions, so permutations are calssified as either even or odd. A poster could give a proof of this non-obvious fact, and show that the set of even permutations in S_n is a subgroup. This is known as the alternating group of degree n. The number of even permutations of n objects is the same as the number of odd permutations, so the alternating group has index 2 in the symmetric group. This fact is not entirely obvious and could form the basis of a poster by itself. See here for information about even and odd permutations and here for alternating groups.
14. Free groups
    For a finite set S with elements x₁,x₂,...,x_k, the free group F(S) on S is a group whose elements are "words" whose characters are the elements of S and their inverses, represented just by symbols. So for example if the elements of S are a and b, the "typical" elements of F(S) are things like ababa^-1 or b²a^-1b³, etc. Whenever a symbol and its inverse appear side by side in a word, we can cancel them and the resulting word represents the same element as the original one. Two words represent the same element if and only if one can be obtained from the other by such steps. Two words represent different elements of the group if they look different. The free group on a single generator is an infinite cyclic group. All free groups on more than one generator are non-abelian. A famous theorem about free groups is the Nielsen-Schreier Theorem, which states that any subgroup of a free group is itself free on some set of generators (possibly with more elements than the generating set of the original group). A counter-intuitive fact about free groups is that a free group with more generators is not "bigger" than a free group on fewer generators - in particular the free group of rank 2 (i.e. on 2 generators) contains copies of the free groups on all finite numbers of generators as subgroups.
15. Presentations and relations
    This is related to the theme of free groups mentioned just above. Suppose that you have a group with two generators, for example the dihedral group of symmetries of the square. Let x and y respectively represent a rotation through 90 degrees and one of the four reflections. Then we know that x and y together generate the group. But they don't generate it freely, because they satisfy some relations, for example x⁴=id and y²=id, also yx=x³y. We can write this group by means of generators and relations as < x,y: x⁴=id, y²=id, yx=x³y> . This is referred to as a presentation of D₈ - it means that D₈ is generated by x and y subject to these relations. That means the elements of D₈ are words in x and y (and their inverses) subject to the rules that we can always replace x⁴ or y² with id, and that we can always replace yx with x³y, enabling us to write all "words" as a power of x followed by a power of y, where the index on the x is in the range 0 to 3, and on the y it is 0 or 1. All groups have descriptions like this - a group with no relations is a free group. Any collection of generators and relations amongst them determines a groups - deciding what kind of properties this group has (even for example whether it is finite or not) is a difficult problem that has inspired advances for example in computational group theory (see below). A poster on this theme could explain the basic concepts of generators and relations and give some examples of presentations of small finite groups, such as the two non-abelian groups of order 8.
16. Computational group theory
    The use of computational techniques to study questions in group theory has been an extremely active area of study in the last 50 years or so, and has yielded new insights into purely algebraic questions as well as developing a life of its own as a research theme. See this wikipedia page for a basic desciption and this survey by Sims for an account of the sorts of problems that can be tackled by computational methods, and of some of the key algorithms that have been devloped.

    Typesetting

    Here is an easy way to introduce yourself to the mathematical typesetting software LaTeX, if you are not already familiar with it. These steps are planned with the poster in mind, but as you will see there you can create all sorts of documents.
    1. Go to Overleaf, which is a browser-based platform designed to support collaboration in LaTeX. Set up a free account for yourself there by clicking "Register".
    2. Click on "New Project" and select "Poster" from the list of templates.
    3. You will see lots of examples there of posters with different designs and layouts. Choose one of them and then click "Open as Template". This will open up your new project, showing you a split screen with the editable .tex code that generates the output on the left, and a view of the .pdf file on the right. The first one, titled "Adversarial Machine Learning" is quite a good one to try first, as everything there is explained quite well in the comments in the .tex file. \\ Any lines that start with the symbol "%" are comments or explanatory notes, they do not influence the output.
    4. You can experiment with the poster text and click "Recompile" at any time to view the updated .pdf output. You can look at the code for organising the content into columns and blocks, emphasizing text in boldface or italics, including files as images, etc.
    5. My advice is just to experiment with one of these templates, adjusting the structure and text to get a sense of what you can do. The material at the top of the .tex file (before "\begin{document}") is setting out the style of the whole document, I suggest leaving that alone for your first experiments.
    6. Here is a template designed for this course, with advice on how to adapt the style, colour scheme etc, and how to typeset some of the most usual mathematical objects. Rather than editing this project directly, please make a copy of it in your own Overleaf account (by clicking on "copy" under "Actions" in your project list) and work with that. Some students have reported difficulty with viewing this Overleaf project. In case you have trouble with it, you can download all the files by clicking on the links below, then add them all to a new Overleaf project in your own account.
       
       • main.tex (The main file with all the content)
       • beamerthemeSharelatex.sty (the style file that specifies the basic layout)
       • tangocolors.sty (Options for colour scheme - see advice in the "colour scheme" section of the main poster)
       • hydrangea.jpg (A picture, as an example)
       • nuigalwaylogo.jpg (The NUI Galway logo, not actually used in this template but potentially useful)
       • ref.bib (The references file)
       
       We will have some time in lectures and tutorials too for getting familiar with LaTeX.

       Practicalities, timeframe etc.

       Identification of groups of two or three should happen by the second week of . Identification of possible topics should -->