In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other.
The set R of real numbers together with the operation of addition and the usual ordering relation between pairs of numbers is an Archimedean group.
The notation na (where n is a natural number) stands for the group sum of n copies of a.
An Archimedean group (G, +, ≤) is a linearly ordered group subject to the following additional condition, the Archimedean property: For every a and b in G which are greater than 0, it is possible to find a natural number n for which the inequality b ≤ na holds.
[3] An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded cyclic subgroups: there does not exist a cyclic subgroup S and an element x with x greater than all elements in S.[4] It is straightforward to see that this is equivalent to the other definition: the Archimedean property for a pair of elements a and b is just the statement that the cyclic subgroup generated by a is not bounded by b.
The sets of the integers, the rational numbers, and the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups.
Conversely, as Otto Hölder showed, every Archimedean group is isomorphic (as an ordered group) to a subgroup of the real numbers.
Let the elements of G be the points of the Euclidean plane, given by their Cartesian coordinates: pairs (x, y) of real numbers.
For every natural number n, it follows from these definitions that n (1, 0) = (n, 0) < (0, 1), so there is no n that satisfies the Archimedean property.
However, there exist non-Archimedean ordered groups with the same property.