Walther von DyckWalther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck (diːk) and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations of combinatorial group theory, being the first to systematically study a group by generators and relations. Von Dyck was a student of Felix Klein, and served as chairman of the commission publishing Klein's encyclopedia. Von Dyck was also the editor of Kepler's works.
Cayley tableNamed after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group - such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center - can be discovered from its Cayley table.
Max DehnMax Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States. Dehn was a student of David Hilbert, and in his habilitation in 1900 Dehn resolved Hilbert's third problem, making him the first to resolve one of Hilbert's well-known 23 problems.
Algebraically closed groupIn group theory, a group is algebraically closed if any finite set of equations and inequations that are applicable to have a solution in without needing a group extension. This notion will be made precise later in the article in . Suppose we wished to find an element of a group satisfying the conditions (equations and inequations): Then it is easy to see that this is impossible because the first two equations imply . In this case we say the set of conditions are inconsistent with .
