Closed Paths in The Coset Diagrams for \(\) Acting on Real Quadratic Fields

Abstract

We are interested in linear-fractional transformations \(y,t\) satisfying the relations \(y^6=t^6 = 1\), with a view to studying an action of the subgroup \(H = \) on \({Q}(\sqrt{n}) \cup \{\infty\}\) by using coset diagrams.

For a fixed non-square positive integer \(n\), if an element \(\alpha = \frac{a+\sqrt {n}}{c}\) and its algebraic conjugate have different signs, then \(\alpha\) is called an ambiguous number. They play an important role in the study of action of the group \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\). In the action of \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\), \(\mathrm{Stab}_\alpha{(H)}\) are the only non-trivial stabilizers and in the orbit \(\alpha H\); there is only one (up to isomorphism). We classify all the ambiguous numbers in the orbit and use this information to see whether the action is transitive or not.