A riddle… -----------
Can you guess which textbooks these two definitions come from?
First definition:
"Let E be a set equipped with an operation denoted by ⋆\star. We say that (E, ⋆\star) is a group if and only if:
1) The binary operation is associative.
2) There is an identity element.
*3) Every element of E has an inverse in E. We say that (E, ⋆\star) is a commutative group if, in addition:*
4) The binary operation is commutative."
Second definition:
"Group structure – A set E has a group structure under a binary operation denoted by ⋆\star if:
1) This operation is associative: (a ⋆\star b) ⋆\star c = a ⋆\star (b ⋆\star c).
2) This operation has an identity element e: a ⋆\star e = e ⋆\star a = a.
3) Every element a has an inverse a’: a ⋆\star a’ = a’ ⋆\star a = e.
A group is said to be commutative (or Abelian) if a ⋆\star b = b ⋆\star a."
From a first- or second-year university textbook? Not even close!
The first comes from a 1973 textbook for the fourth year of collège (eighth grade) (*Mathématique, 4th*, red series, Queysanne-Revuz collection, Fernand Nathan, 1973, 1971 curriculum, page 28).
The second definition comes from a 1967 textbook for final-year classes in the C, D and T streams (Algèbre et Analyse, Lebossé and Hémery collection, Fernand Nathan, 1967, 1966 curriculum, page 26).
Even before "modern mathematics"! ---------------------------------
As early as the 1960s, before the period that would come to be known as "modern mathematics," students in the science tracks were gradually introduced to set-theoretic notions and to group, ring and field structures. In a sense, this brought order to the subject, organizing knowledge to reveal a certain unity within mathematics. It made sense and seemed quite natural to these students in the science tracks.
For example, the curriculum of 18 July 1960 for tenth-grade A’, C, M and M’ classes stated, in the section "‘Modern’ notions: vocabulary and symbolism":
"Other notions, such as those relating to structures on sets—groups, rings and fields—may also be introduced, provided the ground has first been carefully prepared; they can make certain overall presentations easier and allow comparisons that will prove useful in the future."
A short exercise… -----------------
Here is an exercise from a 1971 textbook for the 4th year of French lower secondary school (Mathématiques, classe de quatrième, Monge collection, Belin, 1971, 1971 curriculum, page 44), presented immediately after the definition.
*"Let E = {a, b, c, d}. Consider the operation on PE , denoted by ⋆\star, defined as follows:*
For any two subsets A and B of E, their product is their intersection: A ⋆\star B = A > B.
1) Compute the following: {a} ⋆\star {b}; {a} ⋆\star {a, c}; {a, b, c} ⋆\star {c, d, a}, {b, d} ⋆\star E.
2) Is the operation ⋆\star commutative? Associative?
3) Let A be a subset of E. Compute A ⋆\star E and E ⋆\star A. Show that E is the identity element for the operation ⋆\star.
4) Is there a subset A of E such that {a, b} ⋆\star A = E?
*5) Is the set PE , equipped with the operation ⋆\star, a group?"*
In the fourth year of collège, between 1971 and 1973, the definition of a group was illustrated by the set (Z\Bbb{Z}, +) of integers and by the set (D\Bbb{D}, +) of terminating decimals. In the third year, students learned that all isometries form a group under composition of maps.
Those who received this education may remember it with delight—or despair. The question remains: might it not be wise to reintroduce some structural concepts into secondary education?