Advanced Mathematics

6.2Introduction to the Algebraic Group

Purpose of Section

To introduce the mathematical structure of an algebraic group and to illustrate group concepts, we introduce two important groups: the cyclic and dihedral groups.

6.2.1 Basics of a Group

The idea of a symmetry is fundamental in science and its in‐depth study requires the mathematical machinery of the algebraic group. Group theory is one of the most useful mathematical tools for mathematicians and scientists alike, including particle physicists, where group theory is the language for understanding the structure of particle physics.

Historical Note

The word group was first used by the 20‐year‐old French genius Evariste Galois in 1830, who wrote his seminal paper on the unsolvability of the fifth order polynomial equation on the night before he was killed in a duel. Although many of the greatest mathematicians at the time did not appreciate Galois's brilliance, a letter Galois wrote eventually ended up in the possession of the French mathematician Joseph Liouville, who published Galois's results in 1846. Over the next 50 years, mathematicians gradually came to understand the genius of Galois' ideas and his work gradually developed into the theory of groups.

The most fundamental component of a group is its binary operation.

6.2.2 Binary Operations and the Group

A binary operation on a set A is a rule, which assigns to each pair of elements of A a unique element of A. Thus, a binary operation is a function ...