"A set of symbols 1, α, β,..., all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. [...] These symbols are not in general convertible [commutative] but are associative [...] and it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor [i.e., on the left or on the right], the effect is simply to reproduce the group." (Arthur Cayley, "On the theory of groups, as depending on the symbolic equation θ^n = 1.", 1854)
"This [...] does not in any wise show that the best or easiest mode of treating the general problem is thus to regard it as a problem of substitutions: and it seems clear that the better course is to consider the general problem in itself, and to deduce from it the theory of groups of substitutions." (Arthur Cayley, 1878)
"A group that is not irreducible [indecomposable] can be decomposed into purely irreducible factors. As a rule, such a decomposition can be accomplished in many ways. However, regardless of the way in which it is carried out, the number of irreducible factors is always the same and the factors in the two decompositions can be so paired off that the corresponding factors have the same order." (Georg Frobenius & Ludwig Stickelberger, "On groups of commuting elements", 1879)
"A system G of h arbitrary elements θ1, θ2,...,θh is called a group of degree h if it satisfies the following conditions: (I) By some rule which is designated as composition or multiplication, from any two elements of the same system one derives a new element of the samesystem. In symbols, θrθs = θt. (II) It is always true that (θrθs)θt = θr(θsθt) = θrθsθt. (III) From θθr = θθs or from θrθ = θsθ it follows that θr = θs." (Weber, 1882)
"The following investigations aim to continue the study of the properties of a group in its abstract formulation. In particular, this will pose the question of the extent to which these properties have an invariant character present in all the different realizations of the group, and the question of what leads to the exact determination of their essential group-theoretic content." (Walther von Dyck, "Group-theoretic studies", 1882)
"Now let there be given a sequence of transformations A, B, C,..., . If this sequence has the property that the composite of any two of its transformations yields a transformation that again belongs to the sequence, then the latter will be called a group of transformations." (Felix Klein, 1880s)
"The special subject of group theory extends through all of modern mathematics. As an ordering and classifying principle, it intervenes in the most varied domains." (Felix Klein, 1880s)
"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)
No comments:
Post a Comment