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Edit: I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalent classes of finite groups under the "Isomorphism" equivalent relation. We define a pseudo metric $d$ 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an objects in this completion? Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$? Is there a natural topology on $Z$?

Is $\bar{\mathcal{F}}$ a compact space?

Edit: I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation. We define a pseudo metric $d$ on $\mathcal{F}$ as follows:

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

where $\inf$ is taken over all arbitrary isomorphic copies $\tilde{G}_{n}$ and $\tilde{H}_{n}$ of $G$ and $H$ in $Gl(n,\mathbb{R})$, respectively, while $Hd$ is the Hausdorff distance in $GL(n,\mathbb{R})$ induced by 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an object in this completion? Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$? Is there a natural topology on $Z$?

Is $\bar{\mathcal{F}}$ a compact space?

Edit: I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalent classes of finite groups under the "Isomorphism" equivalent relation. We define a pseudo metric $d$ 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an objects in this completion. Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$? Is there a natural topology on $Z$?

Is $\bar{\mathcal{F}}$ a compact space?

Edit: I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalent classes of finite groups under the "Isomorphism" equivalent relation. We define a pseudo metric $d$ 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an objects in this completion? Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$? Is there a natural topology on $Z$?

Is $\bar{\mathcal{F}}$ a compact space?

**Edit:**I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalent classes of finite groups under the "Isomorphism" equivalent relation. We define a pseudo metric $d$ 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an objects in this completion. Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$?

Is $\bar{\mathcal{F}}$ a compact space?

Edit: I revise the question based on the comment conversations

Let $\mathcal{F}$ be the set of all equivalent classes of finite groups under the "Isomorphism" equivalent relation. We define a pseudo metric $d$ 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 $\bar{\mathcal{F}}$?

Can one consider the unit circle, in some reasonable sense, as an objects in this completion. Is there a natural group structure on every element $Z\in \bar{\mathcal{F}}$? Is there a natural topology on $Z$?

Is $\bar{\mathcal{F}}$ a compact space?