[GAP Forum] ? trouble getting started ?

[Justin Walker]

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
*--------------------------------------*-------------------------------*