# Did Evariste Galois create the entire group structure concept?

Did Evariste Galois create the entire group structure concept?

• If yes, were "super-sets" of groups (e.g. rings or vector spaces) created on top of Galois's work? When and by who?
• If no, did Galois rely on a previously existing concept of groups to create "permutation groups"? Was that concept something fully common/known at that time?

The group concept (and all other important concepts) evolved gradually. Galois did not define the group as we know it now. For him, a group was a group of substitutions on a finite set. Substitutions on finite sets were considered by Cauchy and Lagrange before Galois. Including premutations of the roots of algebraic equations. Galois contribution is that he created a powerful theory (called Galois theory) which uses these permutation groups to analyse algebraic equations. He certainly also had the notion of a field, (domain of rationality) though not formalized in the modern way. The general group concept was formalized during approximately a century (19) with research of Galois and (much later) S. Lie as primary examples of groups.

Same happened with other notions you mention. The notion of vector evolved for almost 2 millenia, beginning with Apollonius, whose famous theorem means exactly that addition of vectors in the plane is commutative. The modern definition of vector space is quite recent (20s century, I suppose) though the ideas which belong to this theory were used by many, including Galois. The development of vector spaces (linear algebra) was not much dependent on the notion of group, because the most interesting examples of groups are non-commutative.

It is important to understand that mathematics did not develop in the way that we learn it in schools and universities.

• So “substitutions on a finite set” mean what we would now call automorphisms of the set? Oct 2, 2015 at 20:31
• Yes. For finite sets, "substitutions" or "permutations" are the terms still used. Oct 2, 2015 at 22:24

Galois's introduced the term group in 1831 in a paper on solutions of polynomials by radicals, but he only considered permutation groups. Without an explicit concept group properties of permutations were used by Lagrange, Ruffini and Cauchy before him. In particular in Lagrange's Reflexions sur la Resolution Algebrique des Equation (1770-71) a theorem on permutations fixing a function is proved, which is now identified with the coset theorem. Cauchy's 1815 paper Memoir on the Number of Values that a Function can Acquire (When One Permutes in all Possible Ways the Quantities it Contains) is considered to be the first paper on permutation groups as such. Galois does not mention Lagrange in 1831, but in his famous last letter he included an example of coset decomposition. He was also likely familiar with Cauchy's work. However, group related notions remained little known until Galois' works were republished in Liouville's Journal in 1846. An abstract definition of (finite) groups appears in Cayley's On the theory of Groups, as Depending on the Symbolic Equation $θ^n = 1$ (1854). See History of Lagrange's Theorem on Groups.

The kind of abstraction that we associate with algebraic structures (as in Bourbaki) did not exist in Galois' time and for some time since. Grassmann introduced (what we now call) vector spaces and exterior algebra in his famous 1844 book, but his approach is in the style of "concrete algebra" and infused with geometry. Dedekind was perhaps first to treat abstract algebraic concepts in their modern spirit, in particular he studied rings of integers and ideals in them. The term number ring was coined by Hilbert in 1892 (published in 1897), and Peano gave the modern definition of vector spaces and linear maps in 1888. Emmy Noether and van der Waerden were largely responsible for promoting the abstract approach before Bourbaki canonized it.

• The article you linked to is very nice! Feb 8, 2016 at 4:58
• Is Lagrange's "Reflexions" available somewhere? Jan 28, 2018 at 15:08
• @BarbaraKwarc Yes, Gallica-Math has a copy scanned from Œuvres Complètes Jan 28, 2018 at 21:48
• Thanks. Is there English translation somewhere too? I don't quite know French :q Jan 30, 2018 at 1:37
• @BarbaraKwarc Apparently, no. MAA's St. George translated only sections 30 and 31 so far. French and German mathematical works are rarely translated into English, it is assumed that one can make out enough between language similarities and formulas. These days one can also put them through Google Translate. Jan 30, 2018 at 3:33