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Re: What difficulties encountered by beginners in abstract algebra?
Posted:
Jan 10, 2013 6:31 PM
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In article
<ken.pledger-F73AB7.10303009012013@news.eternal-september.org>,
Ken Pledger <ken.pledger@vuw.ac.nz> wrote:
> 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, in high school I browsed World of Mathematics.
The article on the group of isometries of the
equilateral triangle showed the normal subgroup
and how to factor it out. It showed how it factors
out in the multiplication table. When I formally
saw group theory this was already in place---a huge help.
--
Michael Press
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