Group, metric, completion Let $G$ be a group, $(G, \rho)$ - metric space, $p: G \rightarrow \mathbb{R}_+$ such that $p(x)=0 \iff x=e_G, \ \ p(x^{-1})=p(x), \ \ p(xy)\le p(x)+p(y), \ \ p(xy)=p(yx)$
Now let $\rho (xy)=p(xy^{-1})$. Properties of $p$ guarantee that $\rho$ is indeed a metric.
Let $(G^-, \rho ^-)$ be the completion of $(G, \rho).$
I need to show that we can define a unique operation $*$ on $G^-$ such that:
1) $(G^-, *)$ is a group an its identity $e = e_G$
2) $x*y=x \cdot y, \ \ x,y \in G$
3) given $q: G^- \ni g \rightarrow \rho ^-(x, e) \in \mathbb{R}_+$ and arbitrary $x,y \in G^-$ $q$ satisfies the last three properties of $p$ and $\rho^-(x,y)= q(x*y^{-1}) = q(x^{-1}*y)$
 A: I don’t clearly understood you, but I expect that my article [1] may be helpful for you, especially, section 3: «The extensions of norms on the closure». 
[1] Ravsky O.V. On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group // Matematychni Studii. – 1999. – {\bf 11}, #1 – P.31-39. 
A: HINT: An element of $\bar G$ is an equivalence class of Cauchy sequences in $G$. Let $x=\langle x_n:n\in\Bbb N\rangle$ and $y=\langle y_n:n\in\Bbb N\rangle$ be Cauchy sequences in $G$; the natural way to try to define $*$ is to set
$$[x]*[y]=[\langle x_n\cdot y_n:n\in\Bbb N\rangle]\;.\tag{1}$$
Of course one has to verify that this is well-defined. Specifically, if $x'=\langle x_n':n\in\Bbb N\rangle$ and $y'=\langle y_n:n\in\Bbb N\rangle$ are Cauchy sequences in $G$ such that $[x']=[x]$ and $[y']=[y]$, one must show that 
$$\lim_{n\to\infty}\rho(x_ny_n,x'_ny'_n)=0\;,$$
which follows (with a little work) from the continuity of the group operation. (You may want to use the fact that $\rho(x,y)=\rho(xy^{-1},e_G)$.) It’s straightforward to verify that $*$ is a group operation on $\bar G$ that agrees with $\cdot$ on $G$, which we identify with its natural embedding in $\bar G$.
Finally, define $q:\bar G\to\Bbb R_+:[x]\mapsto\bar\rho([x],e)$. Suppose that $x=\langle x_n:n\in\Bbb N\rangle$ is a Cauchy sequence in $G$. Then 
$$q([x])=\bar\rho([x],e)=\lim_{n\to\infty}\rho(x_n,e_G)=\lim_{n\to\infty}p(x_n)\;,$$
since for $x_n\in G$ we have $\rho(x_n,e_G)=p(x_ne_G^{-1})=p(x_n)$. You can use this observation to prove that $q$ inherits the properties of $p$.
