Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947.
It is natural to depict cyclically ordered groups as quotients: one has Zn = Z/nZ and T = R/Z.
Even a once-linear group like Z, when bent into a circle, can be thought of as Z2 / Z. Rieger (1946, 1947, 1948) showed that this picture is a generic phenomenon.
For example, Z no longer qualifies, since one has [0, n, −1] for every n. As a corollary to Świerczkowski's proof, every Archimedean cyclically ordered group is a subgroup of T itself.
[3] This result is analogous to Otto Hölder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of R.[4] Every compact cyclically ordered group is a subgroup of T. Gluschankof (1993) showed that a certain subcategory of cyclically ordered groups, the "projectable Ic-groups with weak unit", is equivalent to a certain subcategory of MV-algebras, the "projectable MV-algebras".