symmetry group of the tripod is the smallest group which is not abelian.
Abstract groups
The trend in algebra in the 20th century had been towards abstract algebra, in
which a group is defined by some basic rules known as axioms. With this
viewpoint the symmetry group of the triangle becomes just one example of an
abstract system. There are systems in algebra that are more basic than a group
and require fewer axioms; other systems that are more complex require more
axioms. However the concept of a group is just right and is the most important
algebraic system of all. It is remarkable that from so few axioms such a large
body of knowledge has emerged. The advantage of the abstract method is that
general theorems can be deduced for all groups and applied, if need be, to
specific ones.
A feature of group theory is that there may be small groups sitting inside
bigger ones. The symmetry group of the triskelion of order three is a subgroup
of the symmetry group of the tripod of order six. J.L. Lagrange proved a basic
fact about subgroups. Lagrange’s theorem states that the order of a subgroup
must always divide exactly the order of the group. So we automatically know the
symmetry group of the tripod has no subgroups of order four or five.
Classifying groups
There has been an extensive programme to classify all the possible finite
groups. There is no need to list them all because some groups are built up from
basic ones, and it is the basic ones that are needed. The principle of classification
is much the same as in chemistry where interest is focused on the basic chemical
elements and not the compounds which can be made from them. The symmetry
group of the tripod of six elements is a ‘compound’ being built up from the group
of rotations (of order three) and reflections (of order two).
Axioms for a group
A collection of elements G with ‘multiplication’ ° is called a group if
- There is an element 1 in G so that 1 ° = a ° 1 = a for all elements a in the
group G (the special element 1 is called the identity element).