jats:pWe explore transversals of finite index subgroups of finitely generated groups. We show that whenjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline1" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>is a subgroup of a rank-jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline2" />jats:tex-math$n$</jats:tex-math></jats:alternatives></jats:inline-formula>groupjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline3" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>andjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline4" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>has index at leastjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline5" />jats:tex-math$n$</jats:tex-math></jats:alternatives></jats:inline-formula>injats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline6" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>, we can construct a left transversal forjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline7" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>which contains a generating set of sizejats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline8" />jats:tex-math$n$</jats:tex-math></jats:alternatives></jats:inline-formula>forjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline9" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>; this construction is algorithmic whenjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline10" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>is finitely presented. We also show that, in the case wherejats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline11" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>has rankjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline12" />jats:tex-math$n\le 3$</jats:tex-math></jats:alternatives></jats:inline-formula>, there is a simultaneous left–right transversal forjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline13" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>which contains a generating set of sizejats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline14" />jats:tex-math$n$</jats:tex-math></jats:alternatives></jats:inline-formula>forjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline15" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>. We finish by showing that ifjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline16" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>is a subgroup of a rank-jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline17" />jats:tex-math$n$</jats:tex-math></jats:alternatives></jats:inline-formula>groupjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline18" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>with index less thanjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline19" />jats:tex-math$3\cdot 2n-1$</jats:tex-math></jats:alternatives></jats:inline-formula>, andjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline20" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>contains no primitive elements ofjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline21" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>, thenjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline22" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>is normal injats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline23" />jats:tex-math$G$</jats:tex-math></jats:alternatives></jats:inline-formula>andjats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972715000982_inline24" />jats:tex-math$G/H\cong C2n$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>