Universal valued Abelian groups (Q1941156)

\textit{A value} on a group \(G\) is a function \(p:G\to [0,+\infty)\) such that \(p(x)=0\) if and only if \(x=0\) while \(p(x+y)\leq p(x)+p(y)\) and \(p(-x)=p(x)\) for any \(x,y\in G\). Any value \(p\) on a group \(G\) induces a metric \(d_p\) on \(G\) defined by \(d_p(x,y)=p(x-y)\) for any \(x,y\in G\); let \(\tau_p\) be the topology generated by the metric \(d_p\). \textit{A valued group } is a topological group \(G\) whose topology coincides with \(\tau_p\) for some value \(p\) on \(G\). It is also denoted as a triple \((G,+,p)\). Say that a valued Abelian group \((G,+,p)\) is of class \({\mathcal O}_0\) if \({\displaystyle\lim_{n\to \infty}}{p(na)\over n}=0\) for every \(a\in G\). The symbol \({\mathfrak G}_\infty(0)\) stands for the class of all separable valued Abelian groups of class \({\mathcal O}_0\). The class of separable valued Abelian groups \((G,+,p)\) such that \(p(x,y)\leq 1\) for any \(x,y\in G\) is denoted by \({\mathfrak G}_1(0)\). Observe that \({\mathfrak G}_1(0) \subset {\mathfrak G}_\infty(0)\). Given a natural number \(N>1\), the class \({\mathfrak G}_\infty(N)\) consists of all separable valued Abelian groups \((G,+,p)\) such that \(N\cdot a=0\) for any \(a\in G\) and \({\mathfrak G}_1(N)= {\mathfrak G}_1(0) \cap {\mathfrak G}_\infty(N)\). One of the main results of the paper is the following theorem (Theorem 1.1). \medskip \noindent \textbf{Theorem.} For any \(r\in \{1,\infty\}\) and \(N\in \{0,2,3,4,\ldots\}\) there exists a unique (up to isometric group isomorphism) valued Abelian group \(\mathbb G_r(N)\) with the following properties: \smallskip\noindent \noindent\hangafter=1 \hangindent=.3in\rlap{(g0)}\hskip.3in \(\mathbb G_r(N)\) is complete and belongs to the class \({\mathfrak G}_r(N)\); \noindent\hangafter=1 \hangindent=.3in\rlap{(g1)}\hskip.3in every finite group of class \({\mathfrak G}_r(N)\) admits an isometric group homomorphism into \(\mathbb G_r(N)\); \noindent\hangafter=1 \hangindent=.3in\rlap{(g2)}\hskip.3in every isometric group homomorphism between two finite subgroups of \(\mathbb G_r(N)\) is extendable to an isometric group homomorphism of \(\mathbb G_r(N)\) onto itself; \noindent\hangafter=1 \hangindent=.3in\rlap{(g3)}\hskip.3in the elements of finite order form a dense subset of \(\mathbb G_r(0)\).