I am starting to learn (Visual) Group Theory and I saw this hourglass shaped molecule Ferrocene:
I am wondering what group is described by its symmetries. On top of the five rotations I think there are vertical $v$ and horizontal $h$ flips. Mapping its Cayley Diagram gave me $4$ cycles with flips acting as bridges between them. I've also noticed that $hv = vh$ which is reminiscent of the Klein group $V_4$. Is there a name for this "hourglass group"? And I feel like it could be separated into subcomponents like $V_4$ and a Cycle group of five elements $C_5$? Is there a way to formalize this?
Thanks a lot ! Pierrick
Yes!
The group $G$ should be $D_5\times \Bbb Z_2$, where $D_5$ is the dihedral group of order ten.
If $a$ is a rotation of $2\pi/5$ counterclockwise from a bird's-eye view, $b$ is a horizontal symmetry from that view, and $c$ is a flip about the centre, then we have
$$\begin{align} bab&=a^{-1},\\ a^5&=e,\\ b^2&=e, \\ c^2&=e,\\ ac&=ca,\\ bc&=cb, \end{align}$$
which give the presentation
$$\begin{align} D_5\times \Bbb Z_2&\cong\langle a,b\mid a^5, b^2, bab=a^{-1}\rangle \times\langle c\mid c^2\rangle \\ &\cong\langle a,b, c\mid a^5, b^2, c^2, bab=a^{-1},ac=ca, bc=cb\rangle \end{align}$$
by this standard result.
As we can see here (in particular, here, although it is very technical${}^\dagger$), we can use $\Bbb Z_2^2$ (i.e., $V_4$) and $\Bbb Z_5$, but we would have to use the semidirect product:
$$\boxed{G\cong \Bbb Z_5\rtimes\Bbb Z_2^2.}$$
Also, $G\cong D_{10}.$
$\dagger$: The notation $\Bbb Z_5\rtimes\Bbb Z_2^2$ has some ambiguity. Really, it should be
$$\color{red}{\boxed{G\cong\Bbb Z_5\rtimes_\varphi\Bbb Z_2^2,}}$$
where, if $\mathbb{Z}_5 = \langle x\mid x^5 \rangle$ and $\mathbb{Z}_2^2 = \langle y, z\mid y^2,z^2, yz=zy \rangle$, we have the homomorphism
$$ \varphi:\Bbb Z_2^2\to\Bbb Z_2\le {\rm Aut}(\Bbb Z_5)$$
with $\varphi(y)(x) = x^{-1}$ and $\varphi(z)(x) = x. $ (This determines the whole map by the homomorphism property.)