Abelian groups and the Fundamental Theorem
What do abelian groups look like?
One good way to learn a concept is to see it exemplified, and to see its
lack exemplified. This helps turn a definition that is merely a
collection of words into an idea the student can both picture and remember.
The concept of "abelian group" is an excellent example of the potential this
approach has.
Let's take a look at several Cayley diagrams of abelian groups, and then
several Cayley diagrams for nonabelian groups, then consider what the members
of each separate group have in common.
Cayley diagrams of abelian groups
Cayley diagrams of non-abelian groups
What can we see?
One can notice a few things by careful visual inspection, without knowing
anything about group theory.
 | The abelian groups can be described as "grids" in the sense
that all the lines come together at right angles (except the ones that loop
back). And if we were to chop one of the diagrams up, the lines that
connect the pieces would be parallel, and would connect copies of the same
shape. For example, |
| The nonabelian groups cannot be described as grids, for the reason that
they are more tangled. Lines do not come together at right angles, nor
are they always parallel. In fact, the last of the six images does not
even fully flesh out its rectangular solid shape. |
These two facts make abelian groups seem somehow more predictable or conceptually plainer than non-abelian groups.
Their regular, grid-like nature makes them seem a bit boring.
The nonabelian groups, however, seem to have the potential to get extremely
complicated and tangled. In fact, it's not even immediately obvious
what's going on
in some of the Cayley diagrams above. As sizes of groups increase,
abelian groups will still basically be simple to understand (just giant
grids), whereas nonabelian groups could get
unfathomably tangled.
The Fundamental Theorem of Abelian Groups
These observations are the visual intuition behind the Fundamental
Theorem of Abelian Groups. That theorem says that abelian groups are
easy to understand, because they're always "direct products" (putting together
at right angles) of cyclic (linear) groups.
This is why group theory mainly studies nonabelian groups. The
Fundamental Theorem says all that needs to be said about abelian groups--they're
grids! Having thus completely described every abelian group, the study of group theory
then turns to nonabelian groups, which are much more complex. They're
also generally more interesting:
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