Group theory is the study of mathematical groups, including their symmetries and permutations. It has applications in science, and has become one of the most active branches in all of mathematics in the 20th century.
There are three main sources of group theory. The first source of group theory was number theory, beginning in the late 1700s. A second source was the theory of algebraic equations, leading to the study of permutations, also beginning in the late 1700s. A third source of group theory was geometry beginning around 1800.
Origins of Group Theory
Evariste Galois first coined the term "group theory" in 1830 after he recognized patterns in the roots of quintics. The legend is that he wrote down as many of his developments in this new field as he could by working all night before he was killed, as he expected, in a duel.
Galois certainly fought a duel with Perscheux d'Herbinville on May 30, 1832 (the reason for the duel not being clear but definitely linked with a female; many sources claim she was a prostitute.) Galois was wounded in the duel and was abandoned by d'Herbinville and his own seconds and found later by a peasant. He died in Cochin hospital on the next day, May 31.
In his papers was found a note which reads:
There is something to complete in this demonstration. I do not have the time.
It is this which has led to the legend that he spent his last night writing out all he knew about group theory.