Week 1 Monday: Groups and Group Presentations

Posted on September 14, 2022 by Michaela Polley

Welcome to Geometric Group Theory! As today was the first day of class we began by reviewing the syllabus and discussing expectations for the term before diving into the material.

Review of Groups

Definition: A group is a set with a binary operation G that satisfies three conditions:

  1. G contains an identity element: \exists e\in G such that eg=ge=g for all g\in G.
  2. Every element in G has an inverse: \forall g\in G \exists g^{-1} such that gg^{-1}=g^{-1}g=e
  3. The group operation is associative: \forall g,h,k \in G (gh)k=g(hk)

G can refer to both the set of elements or the group (including the operation). Some authors separate these two uses by denoting the group (G,\cdot). However, in this class we will generally use G to refer to both the set and the group and make clear which one we are referring to.

Let us see this in action. The dihedral group of order 8, called either D_4 or D_8 depending on the text, will be our example. We will call it D_8. The eight elements of the group are:

Since a group is defined by both the set of elements and the group operation, it is not enough to simply list the elements of the group. Instead, we give the “multiplication table” for the group or the Cayley Table. The Cayley Table for the group is shown below.

Group Presentations

A Cayley Table, while helpful, can also be a little bit unwieldy as a way to represent a group. Instead, we give a group presentation, which is a set of generators and relators. Our notation will be G = \langle s_1,\dots,s_n | r_1,\dots,r_k\rangle where each s_i is called a generator and each r_j is called a relator. The set s_1,\dots,s_n is also called the generating set.

Definition: A generating set for G is a set of elements in G such that for all g\in G, g = s_{i_1}^{m_1}s_{i_2}^{m_2}\dots s_{i_n}^{m_n}.

While a generating set may be infinite, each element in G must be the product of a finite number of generators. A product of generators is called a word.

Each of our relators is a word which equals the identity. For example, for D_8.  some relators are V^2, H^2, and (R_{90})^4. Relators can also show the relationship between two elements such that R_{90}^2=R_{180}.

Each group has at least one presentation, since we can always include every element in the group in our generating set. However, this is often not necessary, since we can generate some elements from a set of basis elements. This is similar to finding a basis in linear algebra. For D_8 we know that we don’t need any rotation elements other than R_{90} since each of the other rotations is a power of R_{90}. Similarly, we do not need any flips other than V since we can combine V and R_{90} to attain any of the other flips. Thus, our generating set is \{V, R_{90}\}. By tradition we do not include e in our generating set. For our relators we need to know both how V and R_{90} relate to e and how they relate to each other. Thus, we get V^2, R_{90}^4, and VR_{90}V=R_{90}^{-1} as our set of relators. Therefore, our final group presentation is D_8 = \langle V, R_{90}|V^2,R_{90}^4, VR_{90}V=R_{90}^{-1}\rangle. Next time we will discuss how we know that this is enough to represent the entire group!

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5 Responses to Week 1 Monday: Groups and Group Presentations

  1. Akash Ganguly says:
    September 15, 2022 at 4:31 am

    Super nice Post Michaela! I really liked the Rs you drew in the middle of each square to represent how the shape was being affected after each motion — it made things very clear. Also, I personally appreciated your notes on convention (like not including e in our generating set).

    Reply
  2. Horace Fusco says:
    September 18, 2022 at 7:26 pm

    Wonderful exposition! Whenever I think about group presentations I always think about free groups. These are groups with some given set of generators and no relations between them, so group elements consist of strings of powers of these generators. For example the free group on two generators is <a>, and its elements look like a^{k_1}b^{k_2}a^{k_3} ... b^{k^n} (I’m being a little loose with notation). Free groups can be used to show a lot of interesting things, for instance that subgroups of finitely generated groups are not necessarily finitely generated: take the free group above and consider the subset of words such that k_1 + k_2 + ... + k_n = 0. One can check this is a subgroup, call it H (indeed it is normal, though we won’t use that here). Now for every word a^{k_1}b^{k_2}a^{k_3} ... b^{k^n} \in H, we can ask the following question: what is the maximum value of a partial sum k_1 + ... + k_i for 1 \leq i \leq n. Observe that if H were finitely generated, then this quantity would be bounded for all words in H! (By the maximum such value for each of the finitely many generators). But at the same time we can see that in H this quantity is not bounded, so we are done.

    You can also use free groups to show that there is a group of every cardinality! First note that every finite cardinality has a cyclic group of that order, and second for any infinite cardinality you can take the free group on a generating set X with that cardinality number of generators. To see that this free group has the same cardinality as X, note that for any infinite cardinality, | X \times X | = | X|. Every word in must be a string of finite length, which leaves | X \times X \times X .... \times X | = | X | words of length n, so the total number of words is the union over all n of a set of cardinality | X |, and a countable union of sets of some infinite cardinality again has that same cardinality.

    Reply
  3. Osip Surdutovich says:
    September 19, 2022 at 6:35 am

    Really great post! I really like how you made an effort to show the rotation of the D8 group, it makes it much easier to visualize. Also really clear explanations throughout.

    Reply

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