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.