Arthur Cayley and the Abstract Group Concept

Abstract

In this article, we examine in considerable detail Cayley's first three papers on abstract group theory (1854-59), with special reference to Cayley's formulation of the abstract group concept. We show convincingly that-as far as finite groups are concernedCayley's definition was complete and unequivocal, in contrast to opinion expressed by some other writers. These early papers on abstract group theory [4, 5, 6] seem to have been completely neglected and swept away by the burgeoning subject of permutation groups. The abstract group concept resurfaced in the work of Kronecker in 1870. Arthur Cayley (1821-1895) graduated from Cambridge in 1842 as Senior Wrangler and was awarded the prestigious First Smith's Prize. He then served as Fellow and Tutor of Trinity College (Cambridge) for three years. Since there was no prospect of a permanent academic position, he left for London and entered Lincoln's Inn to qualify for the legal profession. He was called to the Bar in 1849 and he practiced in London as a barrister for the next fourteen years until his appointment as the first incumbent of the Sadleirian Professorship of Pure Mathematics at the University of Cambridge. Even during the years of his legal practice, Cayley published a very large number of research papers in diverse areas of mathematics. Cayley's work spreads over a very wide range of topics, predominantly in the broad fields of algebra and geometry. He was one of the creators of the theory of algebraic invariants. This elaborate edifice, now almost completely forgotten, reigned supreme during the last quarter of the 19th century, and was the "modem higher algebra" of