Arthur Cayley and the ‘Gruppen Pest’

Abstract

The original contributions of Arthur Cayley to the Philosophical Magazine on group theory and his ‘trees’ are revisited and to some extend reinterpreted. Both topics were and are of enormous importance not only in physics (group theory, graph theory), but also in quite a few other disciplines as diverse as information technology or, for example, linguistics (trees, graph theory). In order to show that these two topics originally arose from interests in the theory of permutations also Cayley’s ‘Mousetrap’ game is briefly mentioned.

Keywords:

• history of science

Notes

1 Évariste Galois (1811–1832) was a French mathematician. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel, http://en.wikipedia.org/wiki/Évariste.

2 Let be a homomorphic mapping between groups and , being called the homomorphic mapping of , then is a subgroup of , because

(1)	the identity element , being the identity element in , is the identity element , since , see his definition of identity.
(2)	For follows that , i.e. , see (GT-3(GT-3) ).
(3)	If and , then , since .

3 James Joseph Sylvester FRS (1814–1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. At his death, he was professor at Oxford. Cambridge University denied him for 35 years his B.A. and M.A., because of being Jewish he was not willing to accept the Thirty-Nine Articles of the Church of England. There are quite a few contributions of James Sylvester to be found in the Philosophical Magazine Archives.