Linearly Ordered Subgroup of a Cyclically Ordered Group Which is Not Linear Shely Mutiara Maghfira, Nugroho Dwi Widodo, Rizky Rosjanuardi, Sumanang Muhtar Gozali
Universitas Pendidikan Indonesia, Department of Mathematics Education FPMIPA
Abstract
Every subgroup of a cyclically ordered group \(G\) is cyclically ordered. On a cyclically ordered group which is an external direct product of a linearly ordered group and a cyclically ordered group which is not linear, there could be a subgroup which is cyclically ordered as well as linear. Let \(G\) be a cyclically ordered group which is the external direct product of two cyclically ordered groups which are not linear \(C_{1}\) and \(C_{2}\), written \(G = C_{1}\times C_{2}\). Furthermore, we discuss some condition of a cyclically ordered group \(G\) such that there is a subgroup \(H\) of \(G\) in which the cyclic order on \(H\) is also linear. The condition is by decomposing both of \(C_{1}\) and \(C_{2}\) into the external direct product of a linearly ordered group and a cyclically ordered group which is not linear, suppose that \(C_{1}\)\(=\)\(L_{1}\times C_{11}\) and \(C_{2}\)\(=\)\(L_{2}\times C_{22}\). We take a subgroup \(H\)\(=\)(\(L_{1}\times e_{11} )\)\(\times(L_{2}\times e_{22} )\) of \(G\), \(e_{11}\) is an identity element in \(C_{11}\) and \(e_{22}\) is an identity element in \(C_{22}\).
