One of my favorite theorems in all of group theory is the beautiful Cayley's Theorem, which is named in honor of British mathematician and lawyer Arthur Cayley. Cayley was born #onthisday 202 years ago.

Cayley was the first person to define the concept of a group in the modern (and general) way, that is, as a set with a binary operation satisfying certain laws [1]. Before Cayley when mathematicians talked about "groups" they meant permutation groups.

Cayley had many other mathematical accomplishments. For example, Cayley postulated the Cayley–Hamilton Theorem, which states that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3 [2]. Cayley graphs and Cayley tables are also named in his honor.

A few of my notes on Cayley and Cayley's Theorem are here: https://davidmeyer.github.io/qc/cayleys_theorem.pdf. The LaTeX source is here: https://www.overleaf.com/read/nbfyqkwsfmyc.

As always, questions/comments/corrections/* greatly appreciated.

#math #maths #grouptheory #cayleystheorem

References
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[1] "Arthur Cayley", https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/

[2] "Cayley–Hamilton theorem", https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem