Week 9 - The Dicyclic Groups

Welcome back! Two weeks ago, I explained the approach that Naidu and Rowell took in [1] to prove that the dihedral groups work. As I noted last week, a key reason their proof worked is that all irreducible representations of dihedral groups are either one-dimensional or two-dimensional. This is a very rare property which makes them much easier to work with. However, this property is not unique to them. Another important class of groups, the dicyclic groups, share it.

What are the dicyclic groups?

Dicyclic groups are actually very similar in nature to dihedral groups. This parallel is easier to see from the first of two definitions that I will share.

Dicyclic groups from group presentations

This definition arises from group presentations, which express a group in terms of their generating elements and relations of these generators. I discussed this before in the context of the dihedral group, and I’ll briefly review that now. The dihedral group is generated by a single rotation and by a reflection, which we can call a and b. If the polygon has n sides, then rotating n times should leave no overall change. Similarly, reflecting twice does nothing. We express these conditions as aⁿ=1 and b²=1. Finally, since reflecting changes the orientation of rotations, bab=a⁻¹.

If we work with a polygon that has 2n sides, then the relations will be a²ⁿ=1, b²=1, and bab=a⁻¹.

The presentation of a dicyclic group is almost identical: it is generated by two elements a and b, where a²ⁿ=1, b²=aⁿ, and b⁻¹ab=a⁻¹.

The conjugacy classes of the dicyclic groups

The computation of the conjugacy classes and commutator subgroup of the dicyclic group are quite routine. The conjugacy classes are {1}, {aⁿ}, the pairs of inverses {aᵏ, a⁻ᵏ}, and the classes of {aᵏb} for even and odd k.

These form a total of n+3 conjugacy classes. We can also compute that the commutator subgroup has size 4. Then by the same argument as for the dihedral groups, there are four one-dimensional irreducible representations, and the rest are two-dimensional.

Like for the dihedral groups, one of these one-dimensional irreducible representations is just the trivial representation, sending everything to 1. Another is the sign representation, which sends all rotations to 1 and all reflections to -1. Proceeding by the same arguments, we can derive the same key property: that if X is the sign representation and V is any two-dimensional irreducible representation, then X⊗V=V.

Quaternion groups

Quaternion algebras are a crucial concept used in theoretical physics to better capture three-dimensional structures. They are generated from the real numbers by three terms: i, j, and k. This is an extension of the complex numbers, which are generated by i. The quaternion group captures how these generators interact. Specifically, they are given by the relations i²=j²=k²=-1 and ijk=-1.

The full list of elements is 1, -1, i, -i, j, -j, k, and –k. If we compute the conjugacy classes, we find that they are {1}, {-1}, {i, -i}, {j, -j}, and {k, -k}. This resembles the dicyclic conjugacy classes. If we look more closely, we find that the quaternion group is exactly the dicyclic group with n=2.

This doesn’t help us too much with the general dicyclic groups, but it does show their importance: they are considered generalizations of the quaternion groups.

In fact, we can define dicyclic groups from quaternion algebras: a is a 2n-th root of unity in the usual complex number sense, and b is j. To verify that this is equivalent, we only need to check that the group presentation relations hold.

Other related groups

Before we decide on which other groups we should consider, it’s important to again emphasize why the dicyclic groups are important. Like the dihedral groups, they have a sign representation which is almost always 1, but is -1 for just two conjugacy classes. In general, any one-dimensional representation which is 1 for most conjugacy classes would be very helpful, because by similar arguments, we can try to show that it shares the X⊗V=V property for most V.

In fact, there’s a structural way to interpret this, which I didn’t mention earlier because it’s more technical. Recall that the cyclic group Cₙ is the group of integers modulo n (i.e. adding 1 to itself n times returns us to 0). Then, the dicyclic group is an extension of the cyclic group with 2n elements by the cyclic group with 2 elements.

I won’t formally discuss extensions as they are beyond the scope of this blog, but if you’re interested, they relate to exact sequences.

This also gives us a more intuitive way to think about sign representations. In particular, they focus on the C₂ part of the group. Specifically, it assigns 1 if the component is 0, and -1 if the component is 1.

With this perspective, group extensions (and semidirect product constructions in general) may produce “sign representations”. Accordingly, I’ll focus on these next.

References

[1] D. Naidu, E. Rowell. A finiteness property for braided fusion categories. Algebras and Representation Theory (2011)