# Group theory via actions

I'm teaching undergraduate group theory [again] this term. I've been increasingly dissatisfied with the approach in various books (I've used Fraleigh and looked at others). They're all reasonable books, but I always feel as though much of the material is unmotivated and kind of introduced as "well, here's the next topic".

I started this year by taking some material from the start of Visual Group Theory, showing how object symmetries lead naturally to the definition of a group via arrows in Cayley diagrams. While I like that book, in the end I feel as though it concentrates too much on the visual, stretching it beyond its natural usefulness, and doesn't present enough theory.

Over the past couple of days, I've been thinking about how to continue with the class. The idea of groups arising from symmetries leads naturally to the idea of group actions. So I have in my mind the following sequence of topics and arguments, which constructs most of the basic results using group actions as the basic concept; I'd like comments (mathematic, pedagogical, stylistic, or whatever):

• Introduce group isomorphisms via Cayley diagrams. Discuss homomorphisms and basic properties of kernel and image.
• Introduce group actions as natural extensions of symmetries, prove Cayley embedding theorem, correspondence of actions and homomorphisms $G\to \Omega_X$.
• Discuss conjugation action of $G$ on itself; note that conjugation takes subgroups to [isomorphic] subgroups, note that subgroups that are taken to themselves by every element of $G$ should be worth study; those are [defined to be] normal subgroups.
• Prove that $gHg^{-1} = H \Leftrightarrow gH = Hg$
• Note that kernels are example of normal subgroups.
• Prove that all normal subgroups can be realized as kernels as follows: Given $K\triangleleft G$, we want a map $\varphi$ from $G$ to some other group whose kernel is $K$. That means that if $\varphi(g_1) = \varphi(g_2)$, then $g_1^{-1}g_2\in K$ so that $g_2 = g_1k$. So all elements of the set $gK$ must be taken to the same element of the target group. So lets define the codomain of $\varphi$ to be $\{gK\ |\ g\in G\}$. Then show that there is a group structure on this set exactly when $K$ is normal, and that you get a homomorphism. (Perhaps this all sounds obvious, but to me it completely reverses the normal order of events, which is to introduce normal subgroups as those subgroups whose cosets form a group --- not necessarily an obvious thing to want).
• Discuss quotient groups using the above.
• Use the conjugation action to prove the conjugacy class equation.

At this point I think the students will have seen most of the basic results, assuming that Lagrange and Cauchy and some portion of the abelian group structure theorem are part of the above discussions.

EDIT: what have I done so far?

• Develop diagrams of symmetries of a few standard objects --- a rectangle, an equilateral triangle, and a tetrahedron --- using a minimal set of generators. Here the emphasis is on the symmetry space of the object, not on the operations that induce the rigid motions.
• Point out that while a tetrahedron, a dodecagonal pyramid, and a hexagonal plate each have 12 symmetries, the "relations" between the symmetries are not the same in each case. So for example the tetrahedron has 7 axes of symmetry, the pyramid one, and the plate 7, but the orders of the symmetries are different in the various cases.
• Conclude that to understand symmetries, it's not enough to see how many there are - we have to understand their relationship. Use that to motivate switching focus from the vertices of the graph (the orientations of the objects in space) to the operations themselves; regard the operations as elements, understand what properties they have, and use that to define a group.
• The next section, which I haven't taught yet, is just for basic familiarity: notion of subgroup (using standard arguments --- it's just too hard and not informative to do this visually), extended exploration of cyclic groups, extended exploration of permutation groups. After all of that, I hope to be ready for the material in the first part of this post.
• Generally I like the idea. When asked by first-year students to tell them about higher level maths I will often talk about groups by waving my hands about them acting on things. May I suggest that you include in your question what you've done so far in dot-point form too? That is, make it clear where the definition of groups and subgroups is? It's a little unclear if you want advice for a whole course or just a "how to continue after having started visually". – DavidButlerUofA Aug 29 '14 at 20:44
• @DavidButlerUofA I added some material describing what I've done and what I've definitely planned. – rogerl Aug 29 '14 at 21:37
• I love this approach! I learned group theory out of Herstein's book, which is wonderful but left me with a very abstract approach to the subject. Then I taught it again from a set of notes by Brian Conrad, the first sentence of which—if I remember correctly, "Groups are objects that act on structures"—completely changed the way that I thought about the subject. – LSpice Aug 29 '14 at 22:31
• I believe that I do, though I'm not confident that I could find them. (I know that I have my own notes, but they are very cramped on the page and do not scan well.) I will check on Monday; but, if I do have them, how should I get them to you? – LSpice Aug 30 '14 at 1:00
• This is a question-and-answer site. Will you focus your post with a question, something seeking an answer rather than a discussion? – user173 Aug 30 '14 at 8:34