Let $\mathcal{F}$ be the equivalent classes of all finite groups up to isomorphism. We define a semi metric on $\mathcal{F}$ as follows:

$$d(G,H)= \inf \{Hd(\tilde {G}_{n},\tilde{H}_{n})|\;\} $$

where $\inf$ takes over all arbitrary isomorphic copy $\tilde{G}_{n}$ and $\tilde{H}_{n}$ of $G$ and $H$ in $Gl(n,\mathbb{R})$, respectively. Furthermore $Hd$ is the Hausdorff distance in $GL(n,\mathbb{R})$ based on its standard left invariant metric.

The definition of this metric is motivated by the Hausdorff Gromov metric on the space of compact Riemannian manifolds.

Is $d$ a metric on $\mathcal{F}$? If the answer is yes, we denote by $\bar{\mathcal{F}}$ the completion of $\mathcal{F}$. What can be said about an object $Z$ in $\bar{\mathcal{F}}$? (The Cantor set and the unit circle are two objects in this completion). Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$?