Groups and Group Actions: Lecture 2

In which we meet the dihedral groups, build new groups from old, and explore Cayley tables.

  • Proposition 3: Let G be a group, let g, g_1, g_2, g_3 \in G, let m, n \in \mathbb{Z}.  Then
    1. (g_1 g_2)^{-1} = g_2^{-1} g_1^{-1};
    2. (g^n)^{-1} = (g^{-1})^n;
    3. g^m g^n = g^{m+n};
    4. (g^m)^n = g^{mn};
    5. if g_1 g_2 = g_1 g_3 then g_2 = g_3;
    6. if g_1 g_2 = g_3 g_2 then g_1 = g_3.

    The proof is an exercise in using the group axioms.

  • Definition of a cyclic group.
  • Definition of the nth cyclic group C_n.
  • Definition of the nth dihedral group D_{2n}.
  • Proposition 4: Let P_n be a regular n-gon in the plane.  Write r for the rotation anticlockwise by 2\pi /n about the centre of P_n, and s for the reflection in an axis of P_n.  Then the symmetries of P_n are e, r, r^2, \dotsc, r^{n-1}, s, rs, r^2s, \dotsc, r^{n-1}s. We proved this by labelling the vertices of P_n anticlockwise as 1, 2, \dotsc, n, then picking a symmetry f of P_n, and considering where it sends the vertex 1, splitting into two cases depending on whether the vertices of f(P_n) are numbered anticlockwise or clockwise.
  • Definition of the product group (G\times H, \ast) of two groups (G, \ast_G) and (H, \ast_H).
  • Proposition 5: The operation \ast just defined is a group operation.  This was a straightforward check of the group axioms.
  • Definition of the order of a group, and of a finite group.
  • Definition of a Cayley table.

Understanding today’s lecture

You should have a go at proving the parts of Proposition 3 — I recommend proving enough of the parts that you feel confident that you could prove the rest without difficulty (if necessary, prove all six parts, it’s good practice!).

I encourage you to find the permutations of the vertices of the square corresponding to the eight symmetries we identified.  A good way to check your understanding would be to work out the permutations for the symmetries of the square, and then to check with Richard Earl’s online notes where he gives the permutations (top of page 11).

You could usefully complete the Cayley table for D_6 that we started in lectures, we’ll be thinking a lot about this group so it’s worth investing a little time in familiarising yourself with it now.

You could draw up the Cayley table for some groups we’ve seen so far, eg D_8, or C_n for some sensibly small n like 4 or 5, or \{0, 1\} under addition modulo 2 (that was an example from Lecture 1).  What’s the Cayley table for C_2 \times C_3?  That would be a really good question to explore, it would give you practice with cyclic groups, with a product group and with a Cayley table, but will also link with ideas we’ll meet later in the course.

Further reading

Of course Wikipedia has a page about dihedral groups; it has many pretty pictures.  We talked about Cayley tables today, and also saw that another way to represent a group is via a Cayley graph.  These crop up in various places, for example Fields medallist Terry Tao has written about them on his blog, eg here (warning: this post assumes knowledge of more advanced maths than first-year undergraduates usually have, but you might enjoy skim-reading the post to get a flavour without worrying about understanding it!).  The book Visual Group Theory by Nathan Carter introduces the whole subject of group theory using Cayley graphs and other visualisations, and is well worth a read (you can see sample material and illustrations on that website, and your college library might have a copy).

There are close connections between group theory and the (very English) tradition of change ringing, a particular type of bell ringing.    See for example this NRICH article, which even has a link to online bells so that you can have a go yourself, or this Plus article, which makes a nice connection with the group D_8 of symmetries of a square that we studied today.  Naturally there is an Oxford University society, so if you want to try some practical group theory…

Preparation for Lecture 3

Some of you will want to tackle the first problems sheet before our next lecture, so here are some definitions that you may find helpful.  I’ll give these officially in the lecture too, of course.

  • Definition: Let (G, \ast) be a group.  We say that a subset H \subseteq G is a subgroup of G if the restriction of \ast to H makes H into a group, that is,
    • H is closed under \ast;
    • H has an identity;
    • H contains inverses.
  • Definition: Let G be a group, and take g \in G.  We define the order of g, o(g), to be the smallest positive integer k such that g^k = e.  If no such integer k exists, then we say that g has infinite order.
  • Definition: Let (G, \ast_G) and (H, \ast_H) be two groups.  An isomorphism between G and H is a bijective map \theta : G \to H such that \theta(g_1 \ast_G g_2) = \theta(g_1) \ast_H \theta(g_2) for all g_1, g_2 \in G.  If such an isomorphism exists, then we say that G and H are isomorphic.

And now some questions for you to consider before the next lecture.

Does every Cayley table contain each element exactly once in each row and in each column?

In the definition of a subgroup above, why have I not mentioned associativity?

In the Linear Algebra course, you have studied the structure-preserving maps between vector spaces; these are called linear maps.  In what sense is an isomorphism between groups also an example of a structure-preserving map?  (We shall meet more general structure-preserving maps, which need not always be bijections, later in the course; they are called homomorphisms.)

Last time, I mentioned that composition of functions is a binary operation on the set \mathrm{Sym}(X) of bijections from a set X to itself.  Is \mathrm{Sym}(X) a group under composition of functions?  Is the operation commutative?  If X is a set of size n, what is the size of the set \mathrm{Sym}(X)?

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