Re: Number of cyclic subgroups of order 15 in Z_90 (+) Z_36
- To: mathgroup at smc.vnet.net
- Subject: [mg37994] Re: [mg37986] Number of cyclic subgroups of order 15 in Z_90 (+) Z_36
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Mon, 25 Nov 2002 01:56:16 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
You can use the code I sent you earlier after suitably defining your
group and the multiplication.
Here is the underlying set of pairs:
S = Flatten[Outer[List, Range[0, 89], Range[0, 35]], 1];
Here is the group operation:
mult[{a_, b_}, {c_, d_}] := {Mod[a + c, 90], Mod[b + d, 36]}
This computes the order of an element:
ord[p_] := Length[NestWhileList[mult[p, #1] & , p,
#1 != {0, 0} & ]]
Here is the list of orders, in the same order as the group elements
above. The first number shows the time taken by the computation on my
400 megahertz PowerBook G4.
Map[ord,S]//Timing
{21.26 Second,{1,36,18,12,9,
36,6,36,9,4,18,36,3,36,18,12,9,36,2,36,9,12,18,36,3,36,18,4,9,36,6,
36,9,12,18,36,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,45
,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,30,180,90,60,90,
180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,
90,60,90,180,30,180,90,60,90,180,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,
36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,15,180,
90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,180,45,60,
90,180,15,180,90,60,45,180,30,180,45,60,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,45,180,90,180,45,180,90,180,45,18
0,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,45,180,90,180,10,180,90,60,90,180,30,180,90,20,90,180,
30,180,90,60,90,180,10,180,90,60,90,180,30,180,90,20,90,180,30,
180,90,60,90,180,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,
9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,15,180,90,60,45,180,30,180,45,60,90,180,
15,180,90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,
30,180,45,60,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,6,36,18,
12,18,36,6,36,18,12,18,36,6,36,18,12,18,36,6,36,18,12,18,
36,6,36,18,12,18,36,6,36,18,12,18,36,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,18
0,
90,180,45,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,5,180,90,60,45,180,30,180,45,20,90,180,15,180,90,60,45,180,
10,180,45,60,90,180,15,180,90,20,45,180,30,180,45,60,90,180,90,180,90,18
0,
90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90
,
180,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,
36,18,36,9,36,18,36,9,36,18,36,30,180,90,60,90,180,30,
180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,
90,180,30,180,90,60,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,18
0,\
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,15
,180,\
90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,180,45,60,90,
180,15,180,90,60,45,180,30,180,45,60,90,180,18,36,18,36,18,36,
18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,
18,36,18,36,18,36,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,10,180,90,60,90,180,30,180,90,20,90,180,30,180,90,60,90,180,10,
180,90,60,90,180,30,180,90,20,90,180,30,180,90,60,90,180,45,180,90,180,
45,180,90,180,45,180,90,180,45,180,90,180,
45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,90
,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,3,36,18,12,9,36,
6,36,9,12,18,36,3,36,18,12,9,36,6,36,9,12,18,36,3,36,18,12,9,36,6,36,9,
12,18,36,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,30,
180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,
90,180,30,180,90,60,90,180,30,180,90,60,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,18
0,
45,180,90,180,45,180,90,180,18,36,18,36,18,36,18,36,18,36,18,36,18,36,
18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,5,
180,90,60,45,180,30,180,45,20,90,180,15,180,90,60,45,180,10,
180,45,60,90,180,15,180,90,20,45,180,30,180,45,60,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90
,
180,90,180,90,180,90,180,90,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,18
0,
45,180,90,180,45,180,90,180,30,180,90,60,90,180,30,180,90,60,
90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,18
0,
90,60,90,180,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,
36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,15,180,90,60,45,180,30,180,45,60,90,
180,15,180,90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,
180,45,60,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,2,36,18,12,18,36,6,36,18,4,18,36,6,36,18,12,18,36,2,
36,18,12,18,36,6,36,18,4,18,36,6,36,18,12,18,36,45,180,90,180,45,180,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,45,180,90,180,45,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,15,180,90,60,45,180,30,180,45,60,90,180,
15,180,90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,
30,180,45,60,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,
18,36,9,36,18,36,9,36,18,36,9,36,18,36,30,180,90,60,90,180,30,180,90,
60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30
,
180,90,60,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,5,180,
90,60,45,180,30,180,45,20,90,180,15,180,90,60,45,180,10,180,45,
60,90,180,15,180,90,20,45,180,30,180,45,60,90,180,18,36,
18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,
36,18,36,18,36,18,36,18,36,18,36,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,18
0,
45,180,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,18
0,
30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,45,180,90,18
0,
45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,3,36,18,12,9,36,6,36,9,12,
18,36,3,36,18,12,9,36,6,36,9,12,18,36,3,36,18,12,9,36,6,36,9,12,18,36,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90
,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,45,180,90,180,
45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,
45,180,90,180,45,180,90,180,45,180,90,180,10,180,90,60,90,180,30,180,
90,20,90,180,30,180,90,60,90,180,10,180,90,60,90,180,30,180,90,
20,90,180,30,
180,90,60,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180
,\
45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,18
,36,
18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,
36,18,36,18,36,18,36,18,36,18,36,15,180,90,60,45,180,30,180,45,60,90,
180,15,180,90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,180,4
5,
60,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180
,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,45,180,90
,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,30,180,90,60,90,180,30,
180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,
30,180,90,60,90,180,30,180,90,60,90,180,9,36,18,36,
9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,1
8,
36,9,36,18,36,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,5,180,90,60,45,180,30,180,45,20,90,180,15,180,90,60,45,180,
10,180,45,60,90,180,15,180,90,20,45,180,30,180,45,60,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90
,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,6,36,18,12,18,36,6,36,18,12,18,
36,6,36,18,12,18,36,6,36,18,12,18,36,6,
36,18,12,18,36,6,36,18,12,18,36,45,180,90,180,45,180,90,180,45,180,90,18
0,
45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45
,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,
15,180,90,60,45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,180,45,
60,90,180,15,180,90,60,45,180,30,180,45,60,90,180,90,180,90,180,90,180,9
0,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,18
0,
90,180,90,180,90,180,90,180,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,
9,
36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,9,36,18,36,10,180,90,60,90,180
,
30,180,90,20,90,180,30,180,90,60,90,180,10,180,90,60,90,180,30,180,90,
20,90,180,30,180,90,
60,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180
,\
90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,90,180,90
,180,\
90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,15,180,90,60,
45,180,30,180,45,60,90,180,15,180,90,60,45,180,30,180,45,60,
90,180,15,180,90,60,45,180,30,180,45,60,90,180,18,36,18,36,18,36,18,
36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,36,18,
36,18,36,18,36,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,
180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,30,180,
90,60,90,180,30,180,90,60,90,180,30,180,90,60,90,180,45,180,90,180,45,
180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,45,180,90,180,
45,180,90,180,45,180,90,180,45,180,90,180,90,180,90,180,90,180,90,
180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,180,90,18
0,
90,180,90,180,90,180,90,180}}
On Sunday, November 24, 2002, at 09:15 AM, Diana wrote:
> 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/