Re: Re: Number of cyclic subgroups of order 15 in Z_90 (+) Z_36
- To: mathgroup at smc.vnet.net
- Subject: [mg38020] Re: [mg38002] Re: Number of cyclic subgroups of order 15 in Z_90 (+) Z_36
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Tue, 26 Nov 2002 00:48:47 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
>
> 1. If I want to print out the elements of S which have order 15, do I
> use a
> nested If or Do loop?
I think here by "print out" you mean just make a printable list of the
elements? In that case you do not need any loops or nested Ifs.
Mathematica's programming language is quite different form most others.
It's more like Lisp and Prolog than any more usual ones (you can also
program in that style, called "imperative", but doing so in Mathematica
is inefficient). So here is how to make a list of elements which have
order 15.
S = Flatten[Outer[List, Range[0, 89], Range[0, 35]], 1];
mult[{a_, b_}, {c_, d_}] := {Mod[a + c, 90], Mod[b + d, 36]}
ord[p_] := Length[NestWhileList[mult[p, #1] & , p,
#1 != {0, 0} & ]]
orders = Map[ord,S];
So now S is the list of elements and orders a list of orders. You can
get all elements of order 15 with:
Extract[S,Position[orders,15]]
{{6,0},{6,12},{6,24},{12,0},{12,12},{12,24},{18,12},{18,
24},{24,0},{24,12},{24,24},{36,12},{36,
24},{42,0},{42,12},{42,24},{48,0},{48,12},{48,24},{54,12},{54,
24},{66,0},{66,12},{66,24},{72,12},{72,24},{78,0},{78,12},{78,
24},{84,0},{84,12},{84,24}}
>
> 2. I have just ordered the Mathematica Link to EXCEL package. I will
> use it
> to print out the Cayley Table you e-mailed for A_5. I have not been
> able to
> find a way to get all of the output of the table to print on
> Mathematica. I
> have tried page wrap of various sorts. Is there a way?
I think here by "print out" you mean print out on a printer? This
sounds like a problem with your particular operating system which I
most likely cannot solve. I use Mathematica on Mac OS X and a
PostScript printer any have never had any problems with printing. If
you really have a printing problem you should describe precisely the
hardware and the operating system you are using and someone will
probably be able to help you.
If on the other hand you are referring to "printing out" a table on the
screen: just use the command TableForm to format your output as a
table. It takes various options that can make it look nicer. But I
think you were referring to "real" printing.
With best regards
Andrzej
On Monday, November 25, 2002, at 03:56 PM, Diana wrote:
> Andrzej,
>
> Thanks for your help with this question!
>
> Two questions:
>
> 1. If I want to print out the elements of S which have order 15, do I
> use a
> nested If or Do loop?
>
> 2. I have just ordered the Mathematica Link to EXCEL package. I will
> use it
> to print out the Cayley Table you e-mailed for A_5. I have not been
> able to
> find a way to get all of the output of the table to print on
> Mathematica. I
> have tried page wrap of various sorts. Is there a way?
>
> Thanks for your time, you have made my Abstract Algebra experience so
> much
> more fun.
>
> Diana
>
> "Diana" <diana53xiii at earthlink.remove13.net> wrote in message
> news:arp6l1$1rg$1 at smc.vnet.net...
>> Mathematica groupies,
>>
>> I just got my copy of Mathematica tonight. I understand permutations
>> of
> S_5
>> and A_5, etc., but I am having a little difficulty figuring out how to
>> calculate the order of elements of Z_90 and Z_36, and the order of
> elements
>> of the external direct product of Z_90 (+) Z_36.
>>
>> I am wanting to calculate the number of cyclic subgroups of order 15
>> in
> Z_90
>> (+) Z_36 with Mathematica.
>>
>> Z_90 is the additive group {0, 1, 2, 3, ..., 89}. Z_36 is the additive
> group
>> {0, 1, 2, 3, ..., 36}.
>>
>> So, elements of Z_90 (+) Z_36 would be 2-tuples of the form: (a, b),
>> where
> a
>> is an element of Z_90, and b is an element of Z_36. If you add (a, b)
>> to
>> itself 15 times, you will get {0, 0}. Note that the operation of
>> adding
> the
>> first component of the 2-tuple is modulo 90, and the operation of
>> adding
> the
>> second component of the 2-tuple is modulo 36.
>>
>> Can someone help?
>>
>> Thanks,
>>
>> Diana M.
>>
>>
>>
>
>
>
>
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/