## Timeline of Group Theory

Group theory is about groups of objects that have similar characteristics. Objects can show different kinds of symmetry when they are transformed by a standard set of operations. A basic transformation operation is turning (called rotation in the world of symmetry). When an object is rotated, depending upon the shape of the object and the degree of rotation, the object may appear unchanged or different. A circle can be rotated to any degree and it always appears the same. A square can only be rotated at 90 degree intervals and remain unchanged. Many snowflake patterns can be rotated through 60 degree intervals and remain unchanged. An abstract description of a collection of these symmetries allows objects to be grouped by similarities.

**Group Theory** – [math.uic.edu]

The notion of a group is a vital concept in modern mathematics, and group theory can be thought of as the mathematics of symmetry. The term ‘group’ indicates a group of operations, in which the reverse of each operation is included, and one operation followed by another gives a third operation in the same group. The set of symmetries of an object or pattern always forms a group in this sense, and the group embodies, in an abstract way, the symmetry of the object or pattern concerned.

*timeline:*

1545 – **Cubic equations solved**

1761 – modular arithmetic studied by **Euler**.

1770 – permutations studies by **Lagrange**.

1799 – **Ruffini** claimed that quintic (fifth degree) equations could not be solved algebraically and introduced the idea of groups of permutations.

1801 – **Gauss** expanded on Euler’s work with modular arithmetic, laying the groundwork for theory of Abelian groups.

1815 – **Cauchy** wrote a paper on permutations and their groups.

1824 – **Abel** developed a proof that quintic equations could not be solved.

1831- **Galois** discovered that algebraic solutions of equations are related to groups of permutations.

**An Introduction to Galois Theory** – [maths.org]

1854 – abstract groups – Cayley

1870 – permutation groups – **Jordan** (Camille)

1872 – **Erlangen Program** – **Klein**

1873 – **Lie groups** – **Lie**

1887 – classification of Lie algebras – Killing

1887 – discovery of exceptional group **E-8**

1894 – extension of Killing’s classifications – Cartan

SEE ALSO:

**Introduction to Group Theory** – [dogschool.tripod.com]

**The development of group theory** – [st-andrews.ac.uk]

**Symmetry and Group Theory**