Dear Forum,
I have a finitely presented group
G = fp group on the generators [ f1, f2, x, f3, f4, y ]
(file attached) which I constructed in such a way that:
- < f1, f2, x > is [ 1344, 814 ];
- < f1, f2 > is [ 192, 956 ];
- < f3, f4, y > is [ 576, 8282 ];
- < f3, f4 > is [ 192, 1494 ];
- < f1, f2, f3, f4 > is the Mathieu Group M12;
I would like to identify G somehow, or just know if it the trivial group or
not. I tried to simplify its presentation (also in MAGMA), but it seems it
is not enough. The only thing I know is that it's perfect.
Any advice on how to tackle this kind of problem more efficiently?
Thank you ver much,
William
free:=FreeGroup("f1", "f2", "x", "f3", "f4", "y");;
AssignGeneratorVariables(free);;
g:=free/[ (f2^-1*f1^-1)^2, f2^6, f1^8, f2*f1^-1*f2^-1*f1^2*f2^2*f1^3,
(f2^2*f1*f2^-1*f1^-1)^2, f2*f1^2*f2^-1*f1^2*(f2*f1^-1)^2, f3^3, (f4^-1*f3)^4,
(f3^-1*f4^-1*f3^-1*f4)^2,
f4^8, (f4^2*f3^-1)^2*f4^-2*f3^-1, (f4^2*f3)^3, (f3^-1*f4^-1*f3*f4^2)^2,
(f4^2*f3*f4*f3^-1)^2, (f3*f4^-1*f3^-1*f4^2)^2, (f4^2*f3^-1*f4*f3)^2,
f4^-1*f3*f4*f3^-1*f4^-1*f3*f4^-1*f3^-1*f4*f3*f4^-1*f3^-1,
f1*(f3^-1*f4^-1*f3^-1)^4*(f3^-1*f4*f3^-1)^3,
f2*f1^-1*f2*(f3^-1*f4^-1*f3^-1)^2*f3^-1*f4*f3^-2*f4^-1*f3^-1,
f2^3*(f3^-1*f4*f3^-1)^4*(f3^-1*f4^-1*f3^-1)^2*f3^-1*f4*f3^-1*f4^-1,
(f2^-2*f3)^10, (f2^-2*f4*f3^-1*f4^-1*f3*f4)^10,
(x*f2^-1*(f1^-2*f2*f1^-1)^2*f1^-1*f2^-1*f1^-1)^7, (x*f1^-1*f2*f1^-2)^7,
(x*f1*f2*f1^-4)^8, (x*f2*f1^-2*f2^-1)^3, (x*f1^2*f2^-1*f1)^
7, (x*f1^4*f2^-1*f1^-1)^8, (x*f2*f1^2*f2^-1)^6,
(x*f2^-3*f1^-2*f2*f1^-2*f2^-1*f1^-1)^3, (x*f2^-4*f1^-2*f2*f1^-1)^6,
(x*f2^-3*f1^-3)^6, (x*f1^-3*f2^3)^
3, (x*f2^-4*(f1^-2*f2*f1^-1)^2)^3, (x*f2^-4*f1^-2*f2*f1^2*f2^-1)^6, (x*f2^2)^6,
(x*f2^-1*f1^-2*f2*f1^2*f2^2)^3, (x*f2^-1*f1^-2*f2*f1*f2*f1^-2*f2^-1*f1^-1)^
7, (x*f1^4)^7, (x*f1^-2*f2^-1*f1)^7, (x*f2^-1*f1^-1)^8,
(x*f2^-1*f1^-2*f2*f1^-2)^6, (x*f1*f2^-1*f1^2*f2^2)^7,
(x*f2^-1*(f1^-2*f2*f1^-1)^2*f1^-1)^
8, (x*f1^2*f2^-1*f1^2*f2)^3, (x*f2^-2*(f2^-1*f1)^2*f1*f2)^3,
(x*f2^-3*f1^2*f2*f1^-2*f2^-1*f1^-1)^6, (x*f1*f2^3)^3, (x*f2^-3*f1)^6,
(x*f2^-3*f1^4*f2^-1)^
6, (x*(f1^-1*f2)^2*f2^2)^6, (x*f2^-1*f1^-4*f2^3)^3, (x*f1^2*f2^-1*f1^2*f2^3)^3,
(x*f2^-1*f1^2*f2)^8, (x*f2*f1^2*f2^-1*f1^2)^8, (x*f1^-1*f2)^7, (x*f1*f2*f1^-2)^
6, (x*f2^-1*f1^-1*(f1^-1*f2)^2*f1^-2*f2^-1*f1^-1)^7,
(x*f1*f2^-1*f1^2*f2^2*f1^-2)^7, (x*f1^4*f2^-1*f1)^3, (x*f1^2)^7,
(x*f2^-1*f1^-2*f2)^
8, (x*f1^-2*f2*f1^-2*f2^-1)^8, (x*f1^-1*f2*f1^-4)^7, (x*f2^-1*f1)^6,
(x*f1^-2)^7, (x*f1^2*f2^-1*f1^-1)^7, (x*f2^-1*f1^-2*f2*f1^-3*f2)^
3, (x*f1*f2^-1*f1^2*f2^2*f1^-2*f2^-1*f1^-1)^7,
(x*f2^-4*f1^-2*f2*f1^-3*f2^-1*f1)^8, (x*f1*f2^-1*f1^2)^6, (x*f2^-3*f1^4*f2)^8,
(x*f1^2*f2*f1^-1)^
3, (x*f2^-3*f1^4*f2*f1^-2)^7, (x*f1*f2^-2*f1^-1*(f1^-1*f2)^2*f2^2)^7,
(x*f1*f2^-1)^7, (x*f2^-3*f1*f2^2)^7, (x*f2^-4*f1^-2*f2*f1^-1*f2^-1*f1^-1)^3,
(x*f2*f1^-3)^
7, (x*f2^-3*f1^-2*f2*f1^-2)^6, (x*f1^-1*f2^-1)^7,
(x*f2^-4*f1^-2*f2*f1^-1*f2^-1*f1)^7, (x*f2^-3*f1^-2*f2)^7,
(x*f2^-1*f1^-2*f2*f1*f2^-1)^8, (x*f2*f1^-1)^
8, (x*f2^-3*f1^3)^2, (x*f1^-1*f2^3)^2, (x*f1^-1*f2^-1*f1^2*f2^4)^8,
(x*f2^-3*f1^4*f2*f1^-2*f2^-1*f1^-1)^8, (x*f1*f2^-1*f1*(f1*f2)^2*f2)^
8, (x*f2^-2*(f2^-1*f1)^2*f1*f2^2*f1^-1)^2, (x*f2^-4*f1^-2)^8, (x*f1^2*f2^2)^2,
(x*f2^-2)^8, (x*f1^-2*f2*f1^-1)^3, (x*f2*f1^-4*f2^3)^
8, (x*f2^-1*f1^-2*f2*f1^-1*f2^-1)^6, (x*f2*f1^-2*f2^3)^7, (x*f2^-2*f1^-2)^7,
(x*f2*f1^3*f2^-1*f1^2*f2)^7, (x*f1^5*f2^-1)^7, (x*f2^-3*f1^2*f2*f1^-2)^
3, (x*f1^3*f2^-1)^7, (x*f2^-2*(f2^-1*f1)^2)^6, (x*f2^-3*f1^-1*f2^2)^7,
(x*f2^-3*f1^2*f2)^7, (x*f2^-2*(f2^-1*f1)^2*f1^2)^7, (x*f1^4*f2*f1^-1)^
8, (x*f2^-1*f1^-2*f2*f1^-3*f2^-1)^8, (x*f1^3*f2^3)^4, (x*f2^-3*f1^-1)^4,
(x*f2^-2*f1^-2*f2^-1*f1^-1)^8, (x*f2^-4*f1^-2*f2*f1)^
8, (x*f2^-2*(f2^-1*f1)^2*f1*f2*f1*f2^-1)^8, (x*f2^-3*f1^2*f2^-1)^4,
(x*f2^-1*f1^-2*f2^3)^8, (x*f2^-4*f1^-1*(f1^-1*f2)^2*f1^-1)^4, (x*f2^-3*f1^4)^
7, (x*f2^-4*f1^-2*f2*f1*f2*f1^-2*f2^-1*f1^-1)^7,
(x*f2^-2*(f2^-1*f1)^2*f1*f2^2)^7, (x*f1*f2^4)^8,
(x*f2^-3*f1^4*f2*f1^-2*f2^-1)^6, (x*f2^-3*f1^-2*f2^-1*f1)^
7, (x*f2^-4*f1^-1)^8, (x*f2^-4*f1^-2*f2*f1^-2)^3,
(x*f1^2*f2*f1^-2*f2^-1*f1^-1)^3, (x*f1*f2^-1*f1^2*f2)^6, (x*f1)^3,
(x*f2*f1*(f1*f2^-1)^2*f1^2*f2)^
6, (x*f2^-1*f1^-4)^3, (x*f1^-1*f2*f1^-1)^3, (x*f1^4*f2^-1)^6,
(x*f1^2*f2^-1*f1^2)^6, (x*f2^-4*(f1^-2*f2*f1^-1)^2*f1^-1*f2^-1*f1^-1)^7,
(x*f2^-3)^
7, (x*f2^-3*f1^2*f2^-1*f1)^7, (x*f2^-3*f1^4*f2^-1*f1^-1)^8,
(x*f2^-2*(f2^-1*f1)^2*f1*f2*f1^-1)^3, (x*f1^-1*f2*f1^-2*f2^3)^7,
(x*f2^-4*f1^-2*f2*f1*f2*f1^-2)^
8, (x*f2^-2*f1^-2*f2^-1)^6, (x*f2^-1*f1^-2*f2*f1^-1)^3,
(x*f1^-2*f2*f1^-2*f2^-1*f1^-1)^6, (x*f2^-1*f1^-2*f2*f1*f2*f1^-2*f2^-1)^6,
(x*f1^-3)^3, (x*f2^-1)^
6, (x*f1*f2*f1*f2^-1*f1^2*f2)^3, (x*f2^-1*(f1^-2*f2*f1^-1)^2)^3,
(x*f1^2*f2^-1*f1^-2)^6, (x*f2^-3*f1^-2*f2*f1^-2*f2^-1)^8, (x*f2^-4*f1^-2*f2)^
8, (x*f2^-3*f1^2*f2^-1*f1^-1)^7, (x*f1*f2*f1^-2*f2^3)^3,
(x*f2^-2*(f2^-1*f1)^2*f1*f2^2*f1^-2*f2^-1*f1^-1)^7, (x*f1^-1*f2^4)^7,
(x*f2^-4*f1)^6, (x*f2^-3*f1^-2)^
7, (x*f2^-4*f1^-2*f2*f1^-4)^8, (x*f2^-3*f1^2*f2*f1^-2*f2^-1)^8,
(x*f2^-1*f1^-1*f2^-1*f1^2*f2^4)^7, (x*f1*f2*f1^2*f2^3)^3, (x*f2^-3*f1^2)^
7, (x*f2^-3*f1^-2*f2^-1*f1^-1)^7, (x*f2^-4*f1^-2*f2*f1*f2)^6,
(x*f2^-4*f1^-1*(f1^-1*f2)^2*f1^-2*f2^-1*f1^-1)^7, (x*f2*f1^-4)^8,
(x*f2^-4*f1^-2*f2*f1^-1*f2^-1)^
6, (x*f2)^8, (x*f2^-3*f1^-2*f2*f1^-1)^3, (x*f2*f1^-2)^7, (x*f2*f1^2)^7,
(x*f2^-3*f1^5*f2^-1)^7, (x*f1^5*f2^2)^7, (x*f1*f2^-1*f1)^3, (x*f1^-1*f2^2)^
7, (x*f1^2*f2*f1^-2)^6, (x*f2^-3*f1^3*f2^-1)^7, (x*f1*f2^-1*f1^3)^7,
(x*f1^2*f2)^7, (x*f2*f1^-5*f2^3)^8, (x*f2*f1^-1*f2^3)^8, (x*f1^-1)^
4, (x*f1*f2^-1*f1^2*f2^2*f1^-2*f2^-1)^4, (x*f2^-1*f1^-2*f2*f1)^8,
(x*f2*f1^-2*f2^-1*f1^-1)^8, (x*f2^-1*f1^2)^8,
(x*f2^-1*f1^-1*(f1^-1*f2)^2*f1^-1)^
4, (x*f1^-3*f2*f1^-1)^8, (x*f1^2*f2^-1)^4, (x*f1^4*f2)^8,
(x*f2^-3*f1^2*f2*f1^-1)^3, (x*f2^-1*f1^-2*f2*f1^-3*f2^-1*f1)^8,
(x*f2^-2*(f2^-1*f1)^2*f1)^
6, (x*(f1*f2^-1)^2*f1^2*f2)^7, (x*f1^4*f2*f1^-2)^7, (x*f1*f2^2)^7,
(x*f2^-3*f1*f2^-1)^7, (x*f1^-2*f2*f1^-2)^3, (x*f2^-2*f1)^7,
(x*f2^-1*f1^-2*f2*f1^-1*f2^-1*f1^-1)^
6, (x*f1^3*f2^2)^7, (x*f1^-2*f2)^7, (x*f2^-1*f1^-2*f2*f1^-1*f2^-1*f1)^7,
(x*f2^-2*f1^-1)^8, (x*f2^-2*f1^-5)^8,
(x*f2^-1*f1^-1*(f1^-1*f2)^2*f1^-2*f2^-1)^2, (x*f1^3)^
2, (x*f1*f2*f1^2*f2^-1)^8, (x*f2^-1*f1^-2*f2*f1^-3)^8, (x*f2^-1*f1^-2)^8,
(x*f1^-2*f2^-1)^2, (x*f1^-1*f2*f1*f2^-1*f1^2*f2)^8,
(x*f1*f2^-1*f1^2*f2^2*f1^-1)^2,
y^3, (y^-1 * f4 * f3)^2, f4^2 * y^-1 * f3^-1 * y* f3, f4 * f3 * y * f3^-1 *
y^-1 * f4, (f4^-1 * f3^-1 * y)^2, f4^8, (f4^-1 * f3)^4, f4^-2 * y^-1 * f4^-1 *
f3 * y^-1 * f4^-1 * f3, (f4^-1 * y^-1)^4, (f4^-1 * y^-1 * f4 * y^-1)^2, (f3 *
f4 * f3 * f4^-1)^2, f4 * y * f4^-1 * y^-1 * f3^-1 * f4 * y * f4^-1 * f3, f4^-1
* y^-1 * f4 * y * f3 * f4^-1 * y^-1 * f4 * f3^-1,
y * f4^-1 * y^-1 * f4^-1 * f3^-1 * f4^-1 * f3^-1 * f4^-1 * y * f4^2, f4^-1 * f3
* f4^-1 * y^-1 * f4 * y * f4 * f3^-1 * f4 * y * f4 * f3^-1, y^-1 * f4^-1 * y *
f3^-1 * f4^2 * f3^-1 * y^-1 * f4^-2 * f3^-1 * f4 * f3 * f4^-1 * y^-1 * f3 *
f4^-1, f3^-1 * f4 * y^-1 * f4 * f3^-1 * f4 * y^-1 * f4 * f3^-1 * f4 * y^-1 * f4
* f3^-1 * f4 * f3 * f4^-2 * f3 * f4, (f3 * y)^12];;
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