SECTION 4.2 Basics of Group Theory 207
I have asked this question many times and to many people—some
mathematicians—and often, if not usually, I get the wrong intuitive
response! Without pursuing the details any further, suffice it to say for
now that group theory is the “algebraitization” of symmetry. Put less
obtusely, groups give us a way of “quantifying” symmetry: the larger
the group (which is something we can often compute!) the greater the
symmetry. This is hardly a novel view of group theory. Indeed the
prominent mathematician of the late 19-th and early 20-th century Fe-
lix Klein regarded all of geometry as nothing more than the study of
properties invariant under groups.
Apart from quantifying symmetry, groups can give us a more ex-
plicit way to separate types of symmetry. As we’ll see shortly, the two
geometrical figures below both have four-fold symmetry (that is, they
have groups of order 4), but the nature of the symmetry is different
(the groups are not isomorphic).
Anyway, let’s return briefly to the question raised above, namely
that of the relative symmetry of the two diagrams above. Given a
graph G we now consider the set of all permutations σ of the set of
vertices ofGsuch that
verticesaandbform an edge ofG⇔σ(a) andσ(b) form an edge ofG.A permutation satisfying the above is called an automorphism of
the graphG, and the set of all such automorphisms is often denoted