This is a dated survey of algebraic structures. Despite the title, it does not cover all of mathematics, but just abstract algebra. The skeleton key is an analogy and is explained more clearly in the body as a master key: abstraction in mathematics gives us a way to unlock many problems, just as a master key allows us to unlock many locks, by embodying what they have in common. The present work is a Dover 2002 unaltered reprint of the 1949 Hutchinson & Company edition.
Typically each chapter covers one structure, such as Groups. The chapter gives a strong motivation for the structure (this is the best part of the book), defines it, gives some examples, and outlines a few results. The book is strongest on the oldest structures: groups and fields, along with some number theory. It covers ideals but not rings. It doesn’t mention linear algebra at all, but does have a surprising amount on tensors and invariants.
New algebraic structures get invented all the time, and an obvious weakness of this book is that it can’t cover anything happening after 1949; in fact it stops around the late 1800s with continuous groups, p-adic numbers, and tensors. In particular it does not mention the even more abstract category theory; that would have fit in well with the narrative, except that it wasn’t developed until the 1940s and 1950s.
The book is similar in purpose to the “Guide” sub-series in the MAA’s Dolciani Mathematical Expositions, although the present book only covers a few theorems in each structure while the Guide books go deeper. The present book is pitched low and was probably a popular-math book when it came out, while the Guide books are generally at a graduate level. Birkhoff and Mac Lane’s classic A Survey of Modern Algebra serves somewhat the same purpose and is more accessible for undergraduates.
Bottom line: probably too old and limited to be useful today. It’s too advanced for the general reader, and a student or scholar would be better off with one or more of the concise guides mentioned above.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.