# What group theoretic results were known for several special cases before the general definition of a group was established?

Many results in group theory were proven for permutation groups before the general definition of a group was established (for example: Lagrange's theorem, Sylow's theorems). However, permutation groups weren't the only groups being studied in the nineteenth century, there were also groups of geometric transformations and group arising from number theory (I can't really give more details because I don't frankly know the details).

Were any general group theory results known for several specific cases other than just permutation groups, before the general definition of a group had been formulated? I ask because I'm wondering if such "coincidences" could have motivated the general definition of a group. As an example, Lagrange's theorem was known in the 19th century both for permutation groups and for the multiplicative group of $\mathbb Z/n\mathbb Z$ (via Euler).

• It's probably worth pointing out (though you surely know this) that any purely group theoretic result which can be proven for permutation groups holds for abstract groups by Cayley's theorem. – Logan M Oct 31 '14 at 1:42
• The unsolvability of fifth order polynomials by radicals was proven before Galois as far as I know. He set up the basis for the general case. You could argue how group theoretic this is. – Ali Caglayan Oct 31 '14 at 11:34

The article The Abstract Group Concept, from the McTutor archive, gives an accounting of the steps towards the modern abstract definition. In brief, Cayley made the first stumbling attempts (citing the associative law explicitly) in an 1854 paper, but not until 1895 did Weber give the modern definition, in his Lehrbuch der Algebra. Weber included infinite groups.

As for the original question: other than the Lagrange theorem you mention, I'm not aware of any cases where the abstract definition unified previous separate results. The abstract definition does not seem to have been motivated by this desire. However, it is true that Lie groups were directly inspired by Galois's permutation groups, and Lie's desire to develop a theory for differential equations analogous to Galois theory.

It also seems plausible that Weber, writing a comprehensive text on Algebra, saw the possibility of uniting disparate notions. But that's just a conjecture on my part.

"Almost everything" was found before the general modern definition of a group was extablished:-) I am not sure who gave the first definition of the abstract group (as a set with an operation satisfying such and such axioms). But probably this happened in 20-s century (various people are credited with this). For 19 century mathematicians a group was a group of transformations of a set into itself. And the first deep results belong to Lagrange and Galois.

• Cayley is generally credited with the abstract definition of a group. I would guess in the same 1854 paper where he proved Cayley's theorem. – Michael Weiss Nov 1 '14 at 20:31
• @Michael Weiss: Can you give a reference on Cayley's paper? Did he consider only finite groups or arbitrary ones? If this is so, then all results before 1854 were proven before the general definition of a group. Most notably, Galois theory. – Alexandre Eremenko Nov 1 '14 at 20:39
• books.google.com/…. I haven't read the article myself, hence my phrasing of the comment. – Michael Weiss Nov 1 '14 at 20:41
• @Michal Weiss: well, I read the first page and it confirms what I said: for Cayley the ELEMETNS of the group are "operations", or "transformations", rather the elements of some abstract set:-) I don't think anyone used "sets" systematically before Cantor. – Alexandre Eremenko Nov 1 '14 at 20:46
• Right you are, though if you read the rest of the article you'll find him groping towards the modern concept. Meanwhile I've found the McTutor archive article that outlines the development of the abstract concept. – Michael Weiss Nov 2 '14 at 4:38