Tuesday, January 26, 2016

A conjecture about groups

[Attention, this code is not included in the code site!]

First I want to make a reform, since I want the {}-brackets for sets:

: loc{ [compile] { ; immediate

I have changed this all over.


: { depth >xst ; 
: } depth xst> - negate >set ;













See Mathematics Stack Exchange

The composition of sets of permutations is performed by the word 

pset* ( s1 s2 -- s3 )


: go xst clst ndup >>xst ;

: gen. ( s -- s')  ndup set. nxst@ pset* ;

The word gen. multiply the top set on the data stack with the set on the xst stack and leave the result on the data stack without changing the xst stack. Now the conjecture can be tested.


{ 2143 1234 } { 3412 2143 4321 1234 } pnormal . -1  ok



So the smaller set is a normal subgroup of the bigger set, and therefore the quotient is a group:

{ 3412 2143 4321 1234 } { 2143 1234 }  ok
pquotient set. {{3412,4321},{2143,1234}} ok

This set is a group under pset* and is generated by {3412,4321}:

{ 4321 3412 } go  ok
gen. {4321,3412} ok
gen. {2143,1234} ok
gen. {4321,3412} ok
ndrop  ok

As always with the set elements in quotient groups they have the same number of elements.

Now take a random set of 4-permutations:

{ 2431 2341 } go  ok
gen. {2431,2341} ok
gen. {4132,3142,4312,3412} ok
gen. {1234,1243,3214,4213,1324,1423,3124,4123} ok
gen. {2413,2314,3421,4321,1423,1324,2341,2431,2143,2134,3241,4231, 1243,1234} ok
gen. {4123,3124,4321,3421,2143,2134,4132,3142,4213,3214,4231,3241, 3412,4312,1432,1342,2413,2314,2431,2341} ok
gen. {4231,3241,1234,1243,3214,4213,1432,1342,1324,1423,2134,2143, 2314,2413,4123,3124,4321,3421,4132,3142,4312,3412} ok
gen. {3412,4312,2314,2413,2341,2431,2143,2134,4321,3421,3241,4231, 1342,1432,3142,4132,1234,1243,3214,4213,1324,1423,3124,4123} ok
gen. {4123,3124,3142,4132,3412,4312,1432,1342,3214,4213,2413,2314, 3421,4321,1423,1324,2341,2431,2143,2134,3241,4231,1243,1234} ok
ndrop  ok

In this case the series 

A, A*A, A*A*A,... 

grows until it stagnates in a loop (which is a group), in this case the trivial group generated by the set of the symmetric group of 4-permutations.

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