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Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 8, 2013 4:30 PM
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In article <f49fb6ea-c306-4800-8ad3-7940d8349be6@googlegroups.com>,
Dan Christensen <Dan_Christensen@sympatico.ca> wrote:
> I am working on some introductory notes for group theory. What difficulties
> are typically encountered by math or science undergrads in an introductory
> course on abstract algebra? ....
The abstraction. Too many axioms and proofs one after another can be
not only puzzling but very boring.
When I used to teach a first course in group theory, I made a point
of using lots and lots of examples throughout. Taking it in historical
order I introduced finite permutation groups (needing only the closure
axiom), and gradually worked up to full modern axioms later.
(Actually I started form the Italian solution of the cubic and
quartic, then permutations of the roots, then Lagrange resolvents, then
the permutations leaving such functions invariant; but I'm a history
junkie. :-)
The modern axioms admit infinite groups, of which there are a lot of
examples within the number systems. I used complex roots of unity to
introduce cyclic groups.
Later came groups of isometries leaving invariant various geometrical
figures (e.g. the four-group for the rectangle and rhombus, and why
those two figures must have the same group).
Also Cayley graphs with plenty of colours are good for illustrating
group presentations.
Of course you will come up with different approaches from mine; but
my very strong advice is examples, examples, examples. One of the
beauties of elementary group theory is that there are so many nice
finite examples to look at.
HTH
Ken Pledger.
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