ARCHIMEDEAN GROUP Meaning and
Definition
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An Archimedean group refers to a mathematical structure that combines the properties of a group with an additional property called the Archimedean property. A group, in mathematics, consists of a set of elements and an operation that combines any two elements of the set, satisfying certain axioms. The Archimedean property, on the other hand, is a property associated with the real numbers, stating that for any two positive real numbers, there exists an integer that is larger than the first number, but smaller than the second number.
Thus, an Archimedean group is a group that also possesses the Archimedean property. Specifically, for any two elements in the group, it is possible to find an integer power of one element that is greater than any other given element of the group. This property allows for the comparison and ordering of elements within the group.
Archimedean groups find applications in various areas of mathematics, including algebra, number theory, and analysis. They play a crucial role in understanding certain mathematical structures such as fields, ordered groups, and ordered vector spaces. The Archimedean property ensures that the group has a well-defined order and allows for the generalization of several concepts from the real numbers to these more abstract structures.
In summary, an Archimedean group is a group that satisfies the additional Archimedean property, providing a sense of ordering and comparison among its elements.
Etymology of ARCHIMEDEAN GROUP
The term "Archimedean group" is derived from the name of the Greek mathematician Archimedes (c. 287–212 BCE), who made significant contributions to mathematics. He is known for his discovery of the principle of buoyancy, the approximation of pi, and various geometrical theorems. The term "Archimedean group" specifically refers to a type of mathematical group that satisfies the Archimedean property, which is a property that deals with the order structure of the real numbers. So, the name "Archimedean group" is derived from the association of this mathematical concept with the renowned mathematician Archimedes.
