I considered a Ring-like one dimensional geometry. In this, if we fix an origin (at some point on the circumference), we can think of set of all displacements along the circumference to form a vector space. Now one vector can be denoted by (for some reasons that will become clear), $$ \left( \begin{array}{ccc} x \\ 1 \end{array} \right) $$ Further one can obtain any other vector in the space by translating the vector, say $ x_0 \rightarrow x_0+a $. We can use the linear transformation : $$ T(a) = \left( \begin{array}{cc} 0 & a \\ 0 & 0\end{array} \right) $$ such that $$ \left( \begin{array}{ccc} x + a \\ 1 \end{array} \right) = \left( \begin{array}{ccc} x \\ 1 \end{array} \right)+ T(a)\left( \begin{array}{ccc} x \\ 1 \end{array} \right) $$ Now the set of all such linear transformations will form a group.
Most important part of this transformation is that, if the circumference of the ring is some $L$, then the transformation $T(nL)$ where $ n \in \mathbb Z $ should not change the vector. Mathematically, $$ T(nL) \left( \begin{array}{ccc} x_0 \\ 1 \end{array} \right) = \left( \begin{array}{ccc} x_0 \\ 1 \end{array} \right) $$
Now my question is, with these definitions is the group of Translations a Compact one ? And if it is the generator of the translations will have some properties like angular momenta (although this is a generator of translations) ?
PS : I hope I am not talking about rotations. I am just talking translations along the circumference of the circle.