Background and context regarding the emergence of groups

Our primary sources are Lagrange’s 1770 paper “Réflexions sur la résolution algébrique des équations” and Galois’ Mémoire sur les conditions de résolubilité des équations par radicaux, known as his “first memoir” (hereafter, PM), a manuscript submitted in 1831 but first published posthumously in 1846.Footnote ⁹ It has been established that Galois knew some of Lagrange’s writings (Galois and Neumann Reference Galois and Neumann2011, 5). As such, there is a basis for claiming some intellectual continuity between the two authors.

The mathematical terminology of PM differs from that of “Réflexions.” For the sake of brevity, we will not go into the differences here (see Galois and Neumann Reference Galois and Neumann2011, 18–27; Kiernan Reference Kiernan1971, 40–45). Instead, we simply introduce the following conventions: a “rational expression” in ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ is an expression only in terms of ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ , the four basic arithmetical operations and certain quantities. Precisely which quantities that are allowed depends on the coefficients of the polynomial. To begin with, we may think of these quantities as any number that can be obtained from the coefficients by applying the four arithmetical operations. For example, $3{x_1}{x_2} + {47}{x_3}$ is a rational expression in ${x_1},{x_2}$ and ${x_3}$ . We call the expression in ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ “algebraic” if it is an expression in terms of ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ , the four basic arithmetical operations, the quantities that can be obtained by applying the four arithmetical operations to the coefficients, as well as a finite number of radicals.Footnote ¹⁰ The expression ${x_1}{x_2} + \surd 2{x_3}$ is an example. The radicals are said to be (successively) adjoined to the equation.

In “Réflexions,” Lagrange explores the possibility of expressing the roots of general polynomial equations as algebraic expressions in the coefficients of the polynomial. A polynomial equation for which this is possible is said to be solvable by the adjunction of radicals (solvable, for short). Lagrange was interested in permutations of the roots because he saw a connection between patterns relating to how the roots behave under permutations and the possibility of expressing them algebraically, (i.e., of solving the polynomial). Galois can be said to have continued this line of work. Since permutations will be central in what follows, we briefly outline their relevance in the solvability of polynomials.

In short, in 1797, Carl Friedrich Gauss proved that there are $n$ roots for the general polynomial of degree $n$ (with possible repetitions) but only the general polynomials of degrees 2, 3, and 4 were known to be solvable in the above sense (see, e.g., Kiernan Reference Kiernan1971, 44). Therefore, Lagrange and his contemporaries were interested in figuring out the kinds of roots of different types of polynomials. One way of addressing this question was to search for criteria that guaranteed the existence of solutions of specific kinds of polynomials without having to determine the actual algebraic expressions.

Lagrange’s strategy was to search for a common feature among the techniques for solving the general polynomials of degrees 2, 3, and 4 and then to use this for solving polynomials of a higher degree. One common feature seems to be the introduction of new polynomials constructed in such a way that their roots can be used to find the roots of the original polynomial. The crux is that the roots of these new polynomials are simpler to find because they possess certain symmetry properties with respect to the roots of the original polynomial. This is where permutations enter the investigation of solvability: the roots of the new polynomials must be constructed such that they relate to each other by permutations of the roots of the original polynomial.

The symmetry property of the new polynomials’ roots is a formal relationship between the coefficients and the roots of any polynomial, and it is a key point in Lagrange’s strategy (and later in Galois’ approach, cf. Edwards Reference Edwards1984, 8–9). When some rational expression in the letters $a,b,c,d, \ldots $ is symmetric, in the sense that the expression always remains the same when the letters are permuted among each other, then this symmetric rational expression can be rewritten as a rational expression in the coefficients of the various powers of $x$ that arise from expanding the product $\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)\left( {x - d} \right) \cdots $ . In essence, every symmetric rational expression in the roots of a polynomial can be expressed rationally by its coefficients.Footnote ¹¹

To illustrate the use of permutations in the solution technique, let $p$ be the general polynomial of degree $n$ , and let $f$ be a rational expression in the roots $a,b,c,d, \ldots $ of $p$ . If we permute the roots, the rational expression $f$ will be changed. Now, let ${f_1},{f_2}, \ldots $ be the expressions that all possible permutations of the roots produce from $f$ in this way. Since $p$ has $n$ roots (which we assume to be distinct for simplicity), there are $n!$ permutations of the roots and, thus, $n!$ such permutation variants of $f$ . At least one of them will be equal to $f$ , since one of the permutations leaves every root fixed, but other permutations may also produce expressions equal to $f$ .Footnote ¹² Such relations of symmetry among the ${f_i}$ are used for the construction of the first of the new polynomials mentioned above as follows: we form the polynomial $F\left( x \right) = \left( {x - {f_1}} \right)\left( {x - {f_2}} \right) \cdots \left( {x - {f_{n!}}} \right)$ ; the coefficients of $F$ (i.e., of the various powers of $x$ ) consist in sums and products of the ${f_i}$ and are, thus, rational expressions in the roots of $p$ . Moreover, the coefficients of $F$ are symmetric in the roots of $p$ , i.e., no permutation of the roots of $p$ changes any of the coefficients of $F$ , since any permutation of the roots of $p$ simply changes the order in which ${f_1},{f_2}, \ldots ,{f_{n!}}$ occur in the expression $F\left( x \right)$ above, which does not change $F$ . Therefore, the coefficients of $F$ are rational in the coefficients of $p$ by the symmetry property.

Lagrange’s insight is that the roots of $F$ can be used to determine the roots of $p$ . Obviously, the roots of $F$ are precisely the expressions ${f_1},{f_2}, \ldots $ , but we do not immediately know if they can be expressed algebraically in the coefficients of $p$ . At first glance, this seems like a more challenging problem than that of solving $p$ because the degree of $F$ may be higher than the degree of $p$ . However, the difficulty depends on the relations of symmetry among the ${f_i}$ . If $f$ was chosen wisely, then the ${f_i}$ are not distinct expressions. In this case, it may be possible to rewrite $F$ as a polynomial $G$ of lower degree (with coefficients that are algebraic (if not rational) in the coefficients of $p$ ), thus simplifying the problem. For example, if $p$ is the general quartic (cf. Edwards Reference Edwards1984, §17, 20-21) with roots $a,b,c,d$ , taking $f = \left( {a + b} \right)\left( {c + d} \right)$ gives three different permutation variants ${f_1} = \left( {a + b} \right)\left( {c + d} \right)$ , ${f_2} = \left( {a + c} \right)\left( {b + d} \right)$ and ${f_3} = \left( {a + d} \right)\left( {c + b} \right)$ , each of which arises eight times when the roots $a,b,c,d$ are permuted. This choice of $f$ thus makes it possible to rewrite $F$ in the form ${G^8}$ , where $G$ is a polynomial in ${x^3}$ (to see this, imagine what happens if we expand $F\left( x \right) = {\left( {x - {f_1}} \right)^8}{\left( {x - {f_2}} \right)^8}{\left( {x - {f_3}} \right)^8}$ ). This essentially reduces the problem of finding the roots of $F$ and, therefore, the solution of $p$ to solving a cubic and extracting an 8th root as follows. Since the coefficients of $F$ are rational in the coefficients of $p$ (by the symmetry property), and the coefficients of $G$ are algebraic in those of $F$ ,Footnote ¹³ the coefficients of $G$ are algebraic in those of $p$ . Because $G$ as a cubic is solvable, we known that the roots ${f_1} = \left( {a + b} \right)\left( {c + d} \right)$ , ${f_2} = \left( {a + c} \right)\left( {b + d} \right)$ and ${f_3} = \left( {a + d} \right)\left( {c + b} \right)$ are algebraic in the coefficients of $G$ and, hence, algebraic in the coefficients of $p$ . With some algebraic consideration and manipulation, this eventually leads to the fact that $a,b,c,$ and $d$ are algebraic in the coefficients of $p$ . Thus, with the choice of $f$ , solving the general quartic is transformed into solving a cubic and taking an eighth root. In the general case, we can repeat this technique, if necessary, now rewriting $G$ instead of $F$ , and so on until we reach a polynomial that we can solve. This is why Lagrange and Galois were interested in permutations of the roots of $p$ : which kinds of rational expressions in the roots are changed by some permutations of the roots but not by others? In other words, what are the symmetry properties of rational expressions in the roots of $p$ ? With this short introduction, we now turn to the analysis.

First transformation: Describing permutations as movements

In “Réflexions,” Lagrange simply begins by verbally describing the specific movements of roots, for example, “If we suppose that the roots $x',x'', \ldots ,{x^{\left( \omega \right)}}$ are exchanged respectively into ${x^{\left( {\omega + 1} \right)}},{x^{\left( {\omega + 2} \right)}} \ldots ,{x^{\left( {2\omega } \right)}}$ , or into ${x^{\left( {2\omega + 1} \right)}},{x^{\left( {2\omega + 2} \right)}}, \ldots {x^{\left( {3\omega } \right)}},$ or, etc., it will result in the same changes in the preceding equations as if one exchanged $\zeta '$ into $\zeta ''$ , or into $\zeta '''$ , or, etc.” (Lagrange Reference Lagrange and Serret1869, 323).Footnote ¹⁴ The “preceding equations” that Lagrange mentions are rational expressions in the roots $x',x''$ , etc. (what $\zeta '$ etc. stands for is not important). Importantly, in this quotation, Lagrange explains the permutations by describing how letters denoting roots move with respect to each other (are “exchanged”). The quotation also illustrates that such a description enabled Lagrange to pass smoothly to a description of how the movements of roots affect “the preceding equations”, (i.e., certain given rational expressions in the roots). However, Lagrange is interested in saying something more general about such effects than what can be said by choosing particular explicit expressions in the roots, say $x'x'' + {x^{\left( \omega \right)}}$ , and exchanging its letters. His next step should be seen in light of this interest.

Lagrange introduces a notation for the effects of permutations on expressions as follows: if $f$ is a rational expression in $x$ and $y$ , then $f$ is denoted by $f\left[ {\left( x \right)\left( y \right)} \right]$ . If $f$ remains the same expression when $x$ and $y$ are interchanged, then $f$ is denoted by $f\left[ {\left( {x,y} \right)} \right]$ instead of $f\left[ {\left( x \right)\left( y \right)} \right]$ . For example, we can denote both expressions $xy$ and $2x + y$ by $f\left[ {\left( x \right)\left( y \right)} \right]$ , but only $xy$ can be denoted by $f\left[ {\left( {x,y} \right)} \right]$ . We will call this new notation “the bracket notation” and its expressions, e.g., $f\left[ {\left( x \right)\left( y \right)} \right]$ and $f\left[ {\left( {x,y} \right)} \right]$ , “the bracket expressions.”

Not only does a bracket expression immediately state how permutations affect $f$ , it also makes it easy to compare $f$ to other expressions in the same roots based on how permutations of the roots affect the expressions. Lagrange introduces the convention of using the same bracket expression to denote all rational expressions that are affected in the same way by permutations, and to call such rational expressions similar:

This convention makes the content of the bracket a more important part of a bracket expression than the letter referring to the function (the left-most letter in the bracket expression, such as $f$ and $\phi $ in the quote above). Lagrange thus uses the bracket notation to organize expressions in the roots according to how permutations of the roots affect them.

To describe permutations, Lagrange also begins to use bracket notation instead of always explicating the specific letter movements. For example, rather than writing that $f$ remains the same when $x$ and $y$ are interchanged, he now sometimes writes that $f\left[ {\left( {x,y} \right)} \right]$ is the form of $f$ : “it is necessary, by the hypothesis, that this function also remains the same by changing both $x'$ and $x''$ into $x'''$ and ${x^{IV}}$ ; therefore, since by these permutations the two quantities $y'$ and $y''$ change into each other, the function $f\left[ {\left( {y'} \right)\left( {y''} \right)} \right]$ must be of the form $f\left[ {\left( {y',y''} \right)} \right]$ (Lagrange Reference Lagrange and Serret1869, 359).”Footnote ¹⁶ As a new representation, the bracket notation thus enables Lagrange to describe the effect of a permutation on a rational expression in the roots, without giving a verbal description of how the permutation actually moves the roots.

In addition, notice that there is a subtle difference between $f\left[ {\left( x \right)\left( y \right)} \right]$ and $f\left[ {\left( {x,y} \right)} \right]$ , both materially and referentially. The first bracket expression $f\left[ {\left( x \right)\left( y \right)} \right]$ , which has no comma in it, and with $x$ and $y$ each in their own set of parentheses, simply denotes an unspecified but specifically “chosen” expression $f$ that is rational in $x$ and $y$ —and this may be all we know about $f$ . In contrast, the second bracket expression $f\left[ {\left( {x,y} \right)} \right]$ is sometimes used by Lagrange to say that the unspecified but specifically “chosen” $f$ has a form that is not changed by interchanging $x$ and $y$ ; at other times, this same bracket expression $f\left[ {\left( {x,y} \right)} \right]$ is used to designate this form of symmetry itself. The following passage illustrates the difference: “To find the general form of the function in question, I take … another arbitrary function, designated by $\phi \left[ {\left( {x'} \right)\left( {x''} \right)\left( {x'''} \right)\left( {{x^{IV}}} \right)} \right]$ , and I first reduce it to the form $\phi \left[ {\left( {x',x''} \right)\left( {x''',{x^{IV}}} \right)} \right]$ , so that it remains the same … changing $x'$ to $x''$ or $x'''$ to ${x^{IV}}$ ” (Lagrange Reference Lagrange and Serret1869, 395).Footnote ¹⁷ Thus, in our view, Lagrange’s conceptualization of symmetry as a relation of similarity between functions co-evolves with, and is inseparable from, his ability to capture the idea in a new material configuration, that of the bracket notation. As a new representation, the bracket notation is at once an abstraction and a reification.

The bracket notation states what effect a movement of the roots has on an expression in the roots, even if we do not know what the expression actually looks like (i.e., without knowing specifically how the expression is made up of roots, arithmetical operations, and quantities). This abstraction of the effects of permutations as forms of symmetry makes them independent of one material configuration, namely, a written-out arrangement of the permutable things before and after permutation. But the abstraction comes with a reification, in the form of bracket notation. Although the move may be seen as an abstraction in a traditional sense, the dependence on material configurations does not decrease.

The bracket notation is a form (in the Latourian sense) that takes rational expressions in the roots of a polynomial as matter, similar to the way in which the botanist’s cabinet takes leaves as matter. Just as sorting the leaves in the cabinet creates new possibilities for further work, so does the novel material configuration offered by the bracket notation. In both cases, the organization draws out certain properties (the color of the leaf and the symmetries of the expression) and suppresses others. That the organization works like this is, however, hardly surprising, but once it is complete (the cabinet is full, effects of permutations are expressed in the bracket notation), we also see in the mathematical case that, in the new representation, the forms that the effects of permutations take, and the matter for a subsequent organization, are not clearly distinct.

Second transformation: Tables

The bracket notation is also a central part of the second transformation. As we have seen, Lagrange knew that for a polynomial of degree n, the possibility of obtaining algebraic expressions in the polynomial’s coefficients for its roots depended on the effects of permutations of the roots on some chosen rational expression also found in the roots. As we have also seen, the bracket notation makes it possible to work with the effects of permutations on expressions without committing to a specific explicit expression. The next transformation concerns this use of the bracket notation.

In paragraph 107 of “Réflexions,” Lagrange considers the general polynomial of degree 4, the quartic, after having dealt with the cubic in paragraphs 105–6 (Lagrange Reference Lagrange and Serret1869, 394–397). He denotes the roots $x',x'',x''',{x^{IV}}$ and considers an arbitrary rational expression in the roots without further specifying it. He uses the new bracket notation to identify the expression. Thus, in the bracket notation, the solvability of the quartic now depends on what Lagrange calls the form of $f$ .

Lagrange then lists the $4! = 24$ bracket expressions, which can be obtained by moving $x',x'',x''',{x^{IV}}$ around in $f$ . The list appears in two columns in the original text, as reproduced in Table 1. From here, Lagrange continues as outlined in the background and context section: take a rational expression $f$ in the roots of the general quartic and permute its roots $x',x'',x''',{x^{IV}}$ in all possible ways. This produces $4! = 24$ expressions ${f_1},{f_2}, \ldots ,{f_{24}}$ (one of them is of course $f$ itself). In the bracket notation, $f$ is $f$ , and the 24 bracket expressions listed above denote the 24 permutation variants ${f_i}$ . Lagrange then states:

Table 1. List of bracket expressions (Lagrange Reference Lagrange and Serret1869, 394).

Recall that the possibility of solving the quartic depends on reducing the number of ${f_i}$ , which can be done by choosing $f$ appropriately: we can choose $f$ to be an expression that remains unchanged by some of the twenty-four permutations.Footnote ¹⁹ In Lagrange’s terminology, the ${f_i}$ in the same subset have the same form, (i.e., they are similar). Lagrange’s next move is to assume that any bracket expression in Table 1 is of one of only three forms, namely, $f$ , $f$ or $f$ , which is equivalent to assuming that $f$ has been chosen such that the permutations of the roots $x',x'',x''',{x^{IV}}$ will only produce three different permutation variants of $f$ , in effect reducing the solution of the quartic to the solution of the cubic. In terms of Table 1, this choice corresponds to a subdivision of the twenty-four bracket expressions into three groups of eight.

The novelty in this second transformation is the table. In Table 1, the difference between two bracket expressions shows the movement required to transform one into the other. Notice especially that Table 1 is organized according to the following principle:Footnote ²⁰ the six bracket expressions in the top three-by-two block have the same letter ${x^{IV}}$ in the fourth position, and each of the remaining three three-by-two blocks has a different letter in the fourth position. This means that in order to ensure all twenty-four possibilities get into the table, Lagrange can limit his attention to the first three positions: in each block, he just has to permute the remaining letters (occupying the first three positions) in every possible way. To do so, he permutes the “positions” and, in all the blocks, he does it according to the same pattern, as if the positions were empty. From Table 1’s organization emerges a conceptualization of a permutation that does not require a specific object to move (e.g., a letter) but only positions between which to move. This detachment of the movement itself changes the concept of permutation toward that of an operator that is conceivable without things (e.g., letters) on which to operate.Footnote ²¹ Thus, there seems to be an abstraction of the process of moving letters for the sake of bookkeeping, which detaches the process from the concrete letters. However, this does not move the concept of the permutation further away from materiality since it happens with the material configuration of the table.

The two transformations show that matter and form are in rapid flux. The first transformation organized rational expressions as matter in the form of the bracket notation, and now, in the second transformation, the bracket expressions serve as matter formed by Table 1. Then the table becomes matter with Lagrange’s assumption that it can be divided into three groups of eight similar expressions. Through these transformations, we juxtapose several representations but, in Wagner’s terms (Reference Wagner2019), moving between them does not generate a common invariant independent of the material features of the representations. To return to Latour’s example, when the botanist uses a leaf to represent a location in the forest and uses a particular material aspect of the leaf (color) to place it in the cabinet, she constructs and transforms representations by using the mutability of form and matter, rather than stabilizing a separation of form from matter. In the case of a bracket expression, the feature that makes it represent all rational expressions obtained by certain permutations of the roots is the distribution of parentheses and letters between the brackets. So, when Lagrange compiled the table, he organized the twenty-four bracket expressions according to this same aspect, the distribution of the letters $x',x'',x''',{x^{IV}}$ . Like color, the distribution of the letters does not belong to either form or matter.

Third transformation: Permutations and substitutions

We now pass from Lagrange to Galois. As mentioned earlier, we can assume that Galois draws on some of Lagrange’s work on the use of the systematic interchange of letters to investigate the solvability of equations.Footnote ²² Like Lagrange, Galois introduces various new representations. However, before we consider them, we need to address an important point at which Galois’ terminology diverges from that of Lagrange. Recall that Lagrange uses “permutation” to refer to a re-ordering of the letters that denote the roots. In contrast, Galois uses “permutation” to refer to a string of either single letters, or terms composed of several letters, which denote the roots of some polynomial. The following two examples (only the first is string-like; the second is more row-like, with wide spaces) are what Galois calls permutations (Galois and Neumann Reference Galois and Neumann2011, 125 and 117, respectively).

$$acdb$$

$$\phi V\,\,\,\,\,\, {\phi _1}V\,\,\,\,{\phi _2}V \ldots .. \,\, {\phi _{m - 1}}V$$

In addition, Galois uses the term “substitution,” which Lagrange did not. While Galois does not directly explain what he meant by a “permutation,” he explicitly describes substitutions as the “passage from one permutation to another” (Galois and Neumann Reference Galois and Neumann2011, 115). Thus, in Galois’ use of the terms, “permutation” assumes the sense of a concrete, static inscription. Meanwhile, “substitution” has the sense of the active re-ordering of the individual letters or terms of a permutation, changing it into another permutation (cf. Galois and Neumann Reference Galois and Neumann2011, 20–21). In the following paragraphs, “permutation” and “substitution” refer to Galois’ terminology unless we explicitly state otherwise.

The transformation we now consider involves the relationship between permutations and substitutions. Like Lagrange’s notion of permutation, Galois’ notion of substitution refers to the act of re-ordering letters. But while Lagrange considered these acts relative to expressions in the roots of an equation, Galois, with the distinction between permutations and substitutions, introduces new objects—concrete strings—which provide a new material configuration that allows him to write down substitutions as pairs of permutations. We begin with a passage in which Galois directly addresses this relationship:

In the first paragraph, Galois asserts that substitutions do not depend on any particular permutation: as long as we know how a substitution operates in terms of the chosen letters, it can be represented by taking an arbitrary permutation (i.e., a string of these letters) and re-ordering the letters accordingly to produce another permutation. This pair of permutations then “indicates” the substitution—but so does any other pair of permutations obtained in the same way. For example, the permutations $abc,bac$ together describe the substitution that interchanges $a$ and $b$ and fixes $c$ . This substitution is also described by the pair of permutations $acb,bca$ .Footnote ²⁴

The first paragraph of the quotation thus states an abstraction. Substitutions are abstract in the sense that different pairs of permutations can represent the same substitution. Yet, as the second paragraph makes clear, this abstraction depends on the concrete materiality of permutations. It is impossible to “grasp the idea of a substitution” without having permutations “in the language” (Galois and Neumann Reference Galois and Neumann2011, 115). From our theoretical viewpoint, substitutions are thus introduced as reified in the material configurations of permutations. In Latourian terms, (pairs of) permutations are the matter that displays substitutions as particular forms. Once again, abstraction comes with reification, and form is closely interwoven with matter.

Although Galois seems to attribute a certain primality to permutations in the representation of substitutions, he also includes several other representations of substitutions in PM. One of these is introduced in the latter lines of the quoted passage, with two substitutions denoted by the single capital letters $S$ and $T$ , and the composition of the two substitutions denoted by the concatenation $ST$ .Footnote ²⁵ Thus, Galois introduces a new notation, which he uses to describe the set of substitutions as closed in the sense that when the set contains substitutions $S$ and $T$ , then it contains the substitution $ST$ .Footnote ²⁶

With this capital letter notation, we can designate individual substitutions and, without involving permutations, describe them as compositions of other substitutions. This advances the gradual detachment of substitutions from the material configurations of permutations. However, the chain of abstraction does not take off into pure abstraction since this step of detachment only happens with the reification in the capital letters. At the textual level, a process (that of substituting letters arranged in concrete strings) is turned into a concrete capital letter.Footnote ²⁷ With each tick of abstraction follows a tock of reification.

There is a rich commentary on Galois’ use of capital letter notation. A common point of view is that this kind of notation signals an increased level of abstraction compared with a notation that specifies terms, such as the string of letters (or the tables of strings that we consider in the next transformation). This idea is expressed by Ehrhardt when she considers Galois’ use of capital letters G and H to denote groups.

Here, Ehrhardt suggests that the capital letter notation functions in such a way that its occurrence in PM is central to the development of the mathematical concept of a group. In the context of natural science, this kind of representation has been described as similar to “an object.” Klein, for example, writes, “the fact that a letter is a visible, discrete, and indivisible thing (unlike a written name) constitutes a minimal isomorphy with the postulated object it stands for, namely, the indivisible unit or portion of chemical elements” (Klein Reference Klein2003, 26). In the context of mathematical representations, Grosholz (Reference Grosholz2007) adds (referring to Klein) that, “typographical isolation, for example, is iconic in intent. … I came to similar conclusions [as Klein] about the iconic dimensions of symbols and the symbolic dimensions of icons in mathematics” (Grosholz Reference Grosholz2007, 28). We do not question the validity of these comments, nor the importance of the capital letter notation for the historical development of today’s conceptualization of a group. However, we emphasize that the notation of strings of letters is much more prevalent in PM. In fact, the capital letter notation for substitutions occurs only once in PM (in the passage already quoted). More importantly, however, as we will see in the following transformation, Galois actually uses the strings of letters to prove results about the solvability of polynomials.

Fourth transformation: Groups and subgroups

In the discussion of substitutions, Galois alludes to a practical implication of the independence of substitutions from permutations (Galois and Neumann Reference Galois and Neumann2011, 115). Since a substitution does not depend on any specific pair of permutations, we can produce a representation of a given set of substitutions simply by applying each of the substitutions to a single arbitrary permutation. The list of all the permutations thus obtained (including the initial permutation) describes the group of these substitutions. The next transformation concerns how the permutations that arise from “grouping” substitutions in this way are written. Galois organizes these permutations in rectangular arrangements, as shown in the following passage:

Adjoining this square root to the equation of the 4th degree the group of the equation, which contains 24 substitutions in all, is decomposed into two which contain only 12 of them. Denoting the roots by $a,b,c,d$ , here is one of these groups:

In the following, we refer to this arrangement as Table 2.Footnote ²⁸ Each string of letters is a permutation, and each (ordered) pair of strings from anywhere in this table describes a substitution as the re-ordering of the letters $a,b,c,d$ , required to pass from one string to the other.

By choosing any one of these permutations as the initial permutation and passing from this permutation to each of the others, we obtain twelve substitutions (including the passage from the initial permutation to itself, i.e., the substitution that fixes all the letters). The point is that no matter which permutation we start from, we will get the same twelve substitutions. Equivalently, any substitution determined by a pair of permutations in the table is one of these twelve substitutions.

At this point, the great difference between Table 2 and Table 1 becomes clear. Table 2 is not used for bookkeeping. The twelve permutations are organized in three columns of four, such that each column independently describes a closed set of substitutions. Galois states: “these groups will enjoy the remarkable property that one will pass from one to another by operating on all the permutations of the first with one and the same substitution of letters” (Galois and Neumann Reference Galois and Neumann2011, 119, emphases in original). For example, the substitution that fixes a and moves b to where d is, c to where b is, and d to where c is takes any entry in the first column to the corresponding entry in the second column. Moreover, “the substitutions of letters are the same in each group” (Galois and Neumann Reference Galois and Neumann2011, 121, emphases in original).Footnote ²⁹ Galois does not give this property a name, but for simplicity, we will say that the group “splits.”Footnote ³⁰ The tabular representation thus makes it possible to express splitting as a feature of the organization of the table. Clearly, splitting is more specific than closure: we can easily distort Table 2 such that this feature is lost without changing the (closed) set of substitutions that Table 2 determines.

The splitting of Table 2 is essential when Galois demonstrates that the quartic is solvable.Footnote ³¹ In his demonstration, Galois presents a series of successive transformations of Table 2: first reducing it to its left-most column, then dividing and rearranging this column in a two-by-two table, and finally reducing this resulting two-column table to its left-most column:

$$abcd,$$

$$badc,$$

$$cdab,$$

$$dcab.$$

$$abcd,\,\, cdab,$$

$$badc, \,\,dcab$$

$$abcd,$$

$$badc;$$

The operations “extraction of a single radical of the 3^rd degree” and “extraction of a square root,” with which Galois explains the transformations, refer to adjunctions of a third degree radical and a square root (see the background and context section). The second “extraction of a square root,” after the table has been reduced to the last single column table containing $abcd$ and $badc$ , will reduce this two-permutation table to a single permutation. This successive reduction proves that the general quartic is solvable: “I shall observe to begin with that, to solve an equation, it is necessary to reduce its group successively to the point where it does not contain more than a single permutation. For, when an equation is solved, an arbitrary function of its roots is known, even when it is not invariant under any permutation” (Galois and Neumann Reference Galois and Neumann2011, 121). Galois later adds that this condition is also sufficient (Galois and Neumann Reference Galois and Neumann2011, 125). More importantly, from our point of view, the relationship of the two columns of the middle two-by-two table to each other, and to the whole of this table, is the same as in Table 2, i.e., the group of substitutions of the middle two-by-two table splits just like that of Table 2. Moreover, just like in the case of Table 2, splitting is expressed as a feature of the concrete organization of the table.Footnote ³² The tabular representation makes it possible to show that the general quartic is solvable by successively manipulating a table. Such manipulation occurs on the condition that the table is organized in a way that does not just make splitting a structural property of each of the (smaller and smaller) sets of substitutions, but also a material feature of the organization of the (smaller and smaller) tables.

Harold Edwards has also commented on the structural properties of Galois’ tables. Using the term “presentation” for those tables of permutations that we have called “closed,” Edwards writes, “Galois singled out those subgroups with the property that the various presentations of the subgroup differ from one another by the application of a single substitution” (Edwards Reference Edwards1984, 50). Galois does this without abstracting the splitting property from the tables. For such an abstraction to happen—in our sense of the term—Galois would have to insert yet another representation, such that the move between them would render splitting independent of the tables. Admittedly, Galois does represent property using a different representation, namely the verbal description of how a table that splits looks like, but that description refers too directly to the tables to render splitting an abstract property.Footnote ^{33 ,} Footnote ³⁴

In our interpretation, the material design of Galois’ tabular representation is crucial, as the fundamental property of splitting is reified in the physical layout of the inscription on the paper. Materiality is central, and even a sophisticated structural property—such as splitting—of sets of substitutions is not an abstraction away from material configurations in this case. Rather, splitting emerges as a special organization and reorganization of material configurations. It is a concrete procedure connected to the redistribution of actual inscriptions. However, we cannot say that splitting is strictly a property of the table. This claim can be seen as instantiating Latour’s point that, in dealing with representations, we do not encounter a clear separation between form and matter; the matter constituted by the permutations intertwines with the form constituted by the group. Thus, we see that Galois produces a chain of elements of representation from which an abstract notion of group emerges, yet never ceases to be material.Footnote ³⁵

To recapitulate, we have observed that both Lagrange and Galois investigate the solvability problem using a range of representations. Following the chain of representations from Lagrange’s verbal description of permutations of the roots of polynomials to Galois’ groupings, we did not see a process of even greater detachment from materiality culminating in the abstract concept of mathematical groups. Rather, we saw a series of gradual transformations of representations, with each step involving both an abstraction and a reification in, for instance, the spatial design or typography of a new representation.