[GAP Forum] ? trouble getting started ?
brianbec at comcast.net
brianbec at comcast.net
Sun Jul 25 20:13:08 BST 2004
Thanks all for your generous help. Looks like I just had "conjugation" and "composition" backwards in my mind and I'm now "unstuck" :)
I've got a book by Fraleigh (http://www.amazon.com/exec/obidos/tg/detail/-/0201763907/qid=1090782562/sr=8-1/ref=pd_ka_1/102-1074212-8968942?v=glance&s=books&n=507846 ). I've read through it (all the way lightly) once, and am now looking forward to being able to plow through the exercises using GAP. I had attempted to use Mathematica for that, but I would have had to create a bunch of support for non-commutative multiplication by overloading operators like Circle-Times and what not, or working with matrix representations ALL the time (yucch).
I may be too dumb for algebra, but I'm too smart to try golf <evil grin>
-------------- Original message --------------
> On Jul 25, 2004, at 10:44, Brian Beckman wrote:
> > Hello --
> > I had some trouble understanding Permutations as presented in the
> > tutorial
> > and I wondered whether someone might help me out.
> > I'm working through
> > http://www.gap-system.org/Manuals/doc/htm/tut/CHAP002.htm#SECT008 . I
> > was
> > able to understand "conjugating" permutations with the "caret"
> > operator, so,
> > for instance, (1,2)^(1,2,3)=(2,3); made sense to me and
> > (1,2,3)^(1,2)=(1,3,2); also made sense. I could not figure out
> > "multiplication" of purmutations, however, so (1,2)*(1,2,3)=(1,3); did
> > not
> > make sense to me and (1,2,3)*(1,2)=(2,3); did not make sense. I
> > expected
> > (1,2)*(1,2,3)=(1,2,3)^(1,2) and (1,2,3)*(1,2)=(1,2)^(1,2,3), but that's
> > obviously not the case.
> > I apologize for my ignorance of the subject, but I am attempting to
> > use GAP
> > to learn algebra.
> Maybe an investment in a book on Algebra (like Rotman's, or
> Dummit/Foote) will help :=}.
> > So far, I only know of one kind of operation for
> > permutations (that being composition or conjugation)
> Those (composition, conjugation) are actually two kinds of operation.
> Composition is "apply one, then apply the second", while conjugation is
> two applications of composition: a^b = bab^(-1) (or b^(-1)ab,
> depending on your political party).
> > and I couldn't quickly
> > figure out what your multiplication means.
> Multiplication here is (sort of) composition. If you think of
> multiplication as "apply the left-most first", then your example of
> (1,2)*(1,2,3) works out to be:
> 1 -> 2 -> 3
> 2 -> 1 -> 2
> 3 -> 3 -> 1
> i.e., 2 is left fixed, and 1,3 are transposed, so the result is (1,3).
> In terms of mappings, multiplication in this setting is "composition in
> > I'll continue to play around with
> > it and may possibly find my own answer, but it's humiliating to get
> > frustrated by the very first algebraic operation I attempted here.
> If this is humiliating, don't take up golf :-}.
> FWIW, you will find that mathematics is a discipline and it requires
> its own thought patterns. You will get better with practice, but it
> does take practice.
> Justin C. Walker, Curmudgeon-At-Large *
> Institute for General Semantics | "Weaseling out of things is
> | separates us from the animals.
> | Well, except the weasel."
> | - Homer J Simpson
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