 #### Matt Macauley

08-02-2022

05:26

Can't believe it's already #Week5 in #VisualAlgebra! Today we learned about the dicyclic and diquaternion groups, which aren't widely known (and really hard to motivate w/o visuals), but they are really neat constructions. Definitely tune into this one! 1/14 Mon

But first, we started with permutations matrices. Recall last week, we saw 2 canonical ways to label the Cayley diagrams of S_n with rearrangements of 12....n. Well, these are just left vs. right actions, and can be realized as row vs. column vectors. 2/14 M

Moving on, consider a non-abelian group of order 8. It must have an element of order 4 (exercise), and hence a "partial Cayley diagram" like shown here. There are 2 (non-abelian) ways to complete such a diagram: D_4 and Q_8. 3/14 M

Both of these constructions generalize to groups of order 12. We know D_6, but what does the Q_8 generalization signify? It's called a dicyclic group. (Most people call this Dic_3 or Dic_{12}, but I strongly support Dic_6). You'll see why; it's all about roots of unity. 4/14 M

Dic_6 is what we get if we replace i=e^{2\pi i/4}, a 4th root of unity, in Q_8 with a 6th root of unity. Here are two Cayley diagrams that highlight different features: L: ccw=zeta, inward=j R: ccw=j, inward=zeta 5/14 M

Here, the red nodes are the roots of unity. The blue nodes are the powers of j, and purple is their intersection. Notice how every element of a node on the "vertical axis" has order 4. (Not so clear from the last diagram!) 6/14 M

The cycle diagrams also highlight this structure well. Compare this to the cycle diagrams of D_n (see Week 3). 7/14 M

Now this is key: we can construct a representation of Dic_n by taking the standard 2D-rep of Q_8, and replacing the matrix for i (also the rotation matrix from D_4) with the rotation matrix from D_n. Note: n must be even. We'll explore why on HW 4. 8/14 M

Later in the class*, we'll prove that the quotient of Dic_n by its center, <-1>, is isomorphic to D_{n/2}. Here's a picture of this showing how Dic_6/<-1> ≅ D_3. *Last semester, that was actually on Midterm 2. 9/14 M

Next up: what if we take the quaternion group Q_8=<i,j,k>=<R,S,T>, and throw in the reflection matrix F from D_n=<r,f>=<R,F>? [upper-case means Matrix] We get a group of order 16 that we'll call the diquaternion group. This is a new term as of Dec 2021. 10/14 M

In quantum physics, this is called the "Pauli group on 1 qubit", and generated by the following "Pauli spin matrices": X, Y, Z. We have DQ_n = <X,Y,Z> = <R,S,F>. Both generating sets have their pros & cons. The Q_8 subgroup is in yellow. 11/14 M

The cycle diagrams, with both generating sets, are shown here. Note the 8 elements of order 4, and 6 elements of order 2. Again, yellow nodes are in Q_8. 12/14 M

Recap: 1. We constructed Dic_n from Q_8 by replacing the order-4 rotation matrix R_4 from D_4 with R_n from D_n. 2. We constructed DQ_8 from Q_8 by adding a reflection matrix F from D_n. Let's do BOTH of these! We'll call it a "generalized diquaternion group". 13/14 M

I am not aware of this family of groups being called by another name, other than a central product of C_4 with a dihedral group. Stay tuned for Wednesday, where we construct more fun groups: semidihedral, semiabelian, and learn about semi-direct products! 14/14 M

One way to think about how we constructed Dic_8 was to take D_8 (pic 1), remove the blue arrows (pic 2), and rewire them. But here's another idea. Suppose there's an element s of order 2, not in <r>. Giving us the diagram in pic 3. How can we rewire the RED arrows? 1/11 Wed

One way is to rewire the inner red arrows by the relation srs=r^{n/2-1}. This gives the "semidihedral group", SD_n. Here are the Cayley diagrams of SD_8 and SD_{16}. Do you see another way we can rewire the inner red arrows? 2/11 W

Reversing the inner red arrows defines the relation srs=r^{n/2+1}. A student from last semester suggested the name "semiabelian" for this group, which I love, for a number of reasons. (Stay tuned, or see an earlier thread on SD_n and SA_n.) Here are SA_8 and SA_{16}. 3/11 W

Of course, there's one more re-wiring, defining the abelian group shown here. Its cycle diagram is shown on the right. Remember this diagram, it's gonna come up again really soon... 4/11 W

Here are the cycle diagrams of the 3 other groups that result from rewiring the inner red arrows. One of these should look familiar! One reason why I like "semiabelian" as a name. This is "almost" the abelian group C8 x C2. Same cycle diagram and subgroup lattice! 5/11 W

The matrix representations of these groups are really nice as well. Note that the one not pictured, C8 x C2, results from removing the "bar" from the (2,2)-entry of D_n. 6/11 W

Next up, we revisited direct products, and how their Cayley diagrams have a "box-like shape". We're going to use this to motivate the idea of a semidirect product. 7/11 W

But beware! Sometimes a group is "secretly" a directly product in disguise. Fun trivia: of the following 3 groups, 2 of them are isomorphic to a direct product of non-trivial subgroups. Can you guess which ones? (Hint: it's not the "obvious answer".) 8/11 W

Let's see how to construct a Cayley diagram of a direct product AxB, from Cayley diagrams of A and B. Mnemonic: B is for "Balloon"; we "Blow up" these nodes. A is for "Automorphism" (=rewiring). Let's explain this... 9/11 W

An automorphism is a "structure-preserving rewiring". Here's what we mean by that, by an example. IMPORTANT! There are 2 ways to draw this: (i) edge-rewiring, or (ii) node-relabeling. Sometimes, one of these is more convenient to make the Cayley diagram pretty. 10/11 W

To construct a semidirect product of A with B: Do the "inflation method" to construct AxB, but then stick in re-wired copies of A-Cayley diagrams into the inflated B-nodes. But there are constraints! We'll dig into this more on Friday, so stay with us! 11/11 W

Friday was a FUN way to end a FUN week of #VisualAlgebra! We learned about automorphisms of cyclic groups and how to construct semidirect products A⋊B, where A=C_n. Think of an automorphism as a "structure-preserving rewiring". Here's C_5, as a motivating example. 1/11 Fri

Here's another example, C_7. Notice how the "tripling map" generates Aut(C_7), which is isomorphic to the group U_7. 2/11 F

Here's a Cayley diagram. I wish there was an English word in English....like doubling, tripling, but for a general integer k. (The k-tupling map? Would that work? Any better ideas?) 3/11 F

Moving on, let's construct the 4 semidirect products of A=C_5 with B=C_4, using our visual inflation method (see Wed's thread). Here's the first one. Recall our mnemonic for A⋊B: A = "automorphism" (group getting rewired) B = "balloon" 4/11 F

Here's another semidirect product. Note how the "labeling map" is different. Fun exercise, for next week. Show that one of these groups has an element of order 10 and a unique element of order 2. The other has 5 elements of order 2 and no element of order 10. 5/11 F

There are 2 ways to "render" a Cayley diagram of a rewired group (see thread above from Wed.): re-wire the edges, or re-label the nodes. Here are the 2 semidirect products we constructed, under both conventions. Basically, choose whether to tangle the red or blue edges. 6/11 F

There's another labeling map that works. Exercise: show that this group is isomorphic to one of the first two that we constructed. This shouldn't be too surprising. 7/11 F

There's actually one more. Note that the trivial labeling map yields the direct product. 8/11 F

One concern we should have is whether this construction actually yields a group. Recall the now-infamous Petersen graph example we saw earlier. This is *not* the Cayley diagram of a group! How do we know that our "inflation" construction won't result in such a problem? 9/11 F

On Wednesday, we constructed the semidihedral and semiabelian groups. These are actually direct products of C_n with C_2. In fact, if n=2^m, then there are exactly 4 such semidirect products: D_n, C_n x C_2, SD_n, and SA_n. (Scroll up in this thread for a refresher.) 10/11 F

Here are the constructions of these 4 groups: C_8 x C_2 D_8 SD_8 SA_8 Note: the abelian & semiabelian are the most similar, differing by only 4 blue arrows. Remember (see above, Wed.) they have the same cycle diagrams! Stay tuned for HW 4 this weekend! 11/11 F

Wrapping up the week with a summary of the #VisualAlgebra HW. First problem: here are the four semidirect products of C_8 and C_2. (defined earlier in this thread) Construct each quotient group: {±1,±a,±b,±c,±w,±x,±y,±z} and find the isomorphism type. 1/5 HW 🧵👇

Next up is the diquaternion group. Do the same for this, but also construct a Cayley diagram for generating set DQ_8 = <R,S,F>. Extra credit for the best-looking construction! Yes, I'm shamelessly crowdsourcing this, but looking forward to seeing what they come up with! 2/5 HW

Problem 3: the diquaternion and semidihedral groups, DQ_n & SD_n, are only defined when n=2^m, and the dicyclic group Dic_n when n=2m. But...what groups result if we use the standard presentations or representations for other n? 3/5 HW

Problem 4: construct the Cayley diagram of Aut(C_n) for n=8, 9, 10, 16. On HW 2, they did this for the isomorphic groups U_n (i.e., integers mod n, under mult.) This exercise will reveal WHY these are isomorphic. Scroll up to Friday's posts for an example of Aut(C_7). 4/4 HW

Here is the full pdf: As always, I provide a document of "scratch paper" with things like blank tables and diagrams. Happy Super Bowl Sunday! 5/5 HW 