[GAP Forum] ? trouble getting started ?

[brianbec at comcast.net]

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 
> reverse". 
> 
> > 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. 
> 
> Regards, 
> 
> Justin 
> 
> -- 
> Justin C. Walker, Curmudgeon-At-Large * 
> Institute for General Semantics | "Weaseling out of things is 
> what 
> | separates us from the animals. 
> | Well, except the weasel." 
> | - Homer J Simpson 
> *--------------------------------------*-------------------------------* 
>