## In search of an isomorphism

I’m working on some group-theoretic stuff related to n copies of a unital quantum channel and am in search of some isomorphism between the dihedral group $\textrm{D}_n$ and the unitary group U(n).

Now, what’s interesting is that the unitary group U(1) is isomorphic to the circle group, i.e. the geometric circle, while $\textrm{D}_n$ can be thought of as representing the discrete rotations of an n-gon. So one would think that as $n \to \infty$, $\textrm{D}_n$ would become isomorphic to U(1) since the n-gon geometrically approaches a circle. But $\textrm{D}_\infty$ is isomorphic to $\mathbb{Z}$ which geometrically is a line infinite in both directions.

Technically, for what I’m interested in, I suspect U(1) would be too restrictive anyway.  I’m more interested in finding a direct isomorphism between $\textrm{D}_n$ and U(n) or, at the very least, something broader than U(1) and, preferably, broader than SU(2) as well.

So if you’re a regular reader of this blog and you know of any such isomorphism, post it here!

### 3 Responses to “In search of an isomorphism”

1. Pete L. Clark Says:

Unfortunately the symbols with generators and relations disappeared.

In plainer text, D_n is generated by elements a and b and subject to the relation that a^n = b^2 = 1, b a b^{-1} = a^{-1}, and D{\infty} has the same generating set but omits the relation a^n = 1.

2. quantummoxie Says:

Pete,

Huh. Wonder where I read that (about $D_{\infty}$ being isomorphic to $\mathbb{Z}$). Thanks for the clarification.