Group theory in pedagogy
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This article describes how group theory is taught in the classroom, at various levels of mathematics education.
In high school
Group theory is not universally treated as part of the high school math curriculum, though there are schools and curricula where some preliminary notions of group theory are introduced.
Although group theory is not directly addressed in mathematics Olympiads, there are areas of Olympiad mathematics that are closely related to the theory of groups, particularly for Abelian groups. Particular examples:
- Number theory: The theory of the ring of integers modulo , and in particular the field of prime numbers, relates to ideas of Abelian group theory. For instance, Fermat's little theorem can be viewed as a corollary of Lagrange's theorem.
- Geometry: Some glimpses of group theory are seen in transformation geometry; in particular, one sees the group of Euclidean motions as rotations, translations and reflections. The holistic structure of the group is not studied much but individual transformations are often used as tools to solve problems. Many of the groups encountered here are non-Abelian.
- Combinatorics: Some ideas of group theory are used for refined counting arguments involving situations of symmetry. This is the first encounter with the symmetric groups and alternating groups. For instance, Polya's enumeration techniques are sometimes taught as part of Olympiad preparation. Often, the groups encountered here are non-Abelian.
- Algebra: Some of the ideas in solving functional equations, use the ideas of groups and homomorphisms.
In physics and chemistry
Some of the ideas of invariance and symmetry are used in physics and chemistry. Symmetry-invariance is used as a tool to compute integrals that arise in electrodynamics. For instance, the rotational symmetry of a situation helps us conclude that there is no force in a particular direction.
High school chemistry also encounters some ideas of groups and symmetries; this comes up a bit in stereochemistry, in enumerating lattice structures. However, explicit use of group theory is not usually made.
In undergraduate coursework
Group theory course
Group theory is often covered at two stages in the college curricula:
- Along with linear algebra, group theory is part of the first introduction to algebra. In a first course, students are usually exposed to the notions of group, Abelian group, subgroup, coset, Lagrange's theorem, normal subgroup and group homomorphism and direct products.
- At a later stage, students study the theory of group actions, along with common ideas like Sylow's theorems and the Sylow structure of groups, a bit about nilpotent groups and solvable groups, the notion of a simple group and a composition series. The second course varies from place to place and is not very standard.
Use of group theory in other courses
One of the reasons why group theory is taught towards the beginning of algebra coursework is that the notion and language of groups is used throughout algebra, although very few further advanced results of group theory are required.
- Linear algebra: Groups turn up in linear algebra in two forms: as additive groups of vector spaces, and as multiplicative groups of linear transformations (like the general linear group, orthogonal group, and symplectic group). Some ideas from group theory help put in perspective results in linear algebra, even though they can often be stated without a knowledge of group theory.
- Topology: Non-Abelian groups turn up here in the form of the fundamental group, and we encouter ideas of normal subgroups and homomorphisms as well. Groups also turn up in the form of topological groups.
- Differential geometry: Here, groups turn up again in two forms: as Lie groups, and as the groups in which transition functions across coordinate charts live. Most of the groups that turn up here are linear (i.e. subgroups of the general linear group).
- Commutative algebra: Groups turn up here in a minimal fashion: the underlying additive structure of a commutative ring is an Abelian group. They also turn up by analogy: a lot of the theory of ideals in commutative rings is analogous to the theory of normal subgroups in groups.
- Discrete mathematics and combinatorics: Groups occur here as symmetries.
- Linear representation theory: A course in linear representation theory comes back again to the structure of finite groups, and revives the idea of normal subgroups, inner automorphisms and conjugacy classes. However, not too much in-depth knowledge of groups is required to understand the proofs of the main results.
- Galois theory: This again uses the ideas of groups, normal subgroups, conjugate subgroups, and group homomorphisms.
Group theory as a test case
Groups are often treated as a first case for algebraic structures. Thus, in a number of cases, the first example given is that of a group. To introduce the idea of a category, one may begin by studying the category of groups, or to introduce the ideas of universal algebra, one may begin by looking at the variety of groups.
There is a growing divide between group theory and mathematics. People outside group theory often are unaware of the extent to which ideas in group theory have been developed and the role that these ideas play in other subjects. Part of the reason may be that in undergraduate education, very few of the ideas from group theory beyond the basic definitions are applied systematically to other subjects. Thus, at a stage where the mathematical preferences of students are being formed, they may not appreciate the importance of ideas in groups.