Well.. It is known that the groups
M(a,b,c)=<x,y,z | x^y=x^a, y^z=y^b, z^x=z^b> are finite (see for example
D.L. Johnson Presentations of groups, Ch.7, exercise 16).
(Here a,b,c are natural numbers, x^y=y^(-1)*x*y.)
But running
GAP (inside SAGE) I have got:
sage: f:=FreeGroup("a","b","c");
sage:
G:=f/[f.2^-1*f.1*f.2*f.1^-4,f.3^-1*f.2*f.3*f.2^-4,f.1^-1*f.3*f.1*f.3^-4];
<free group on the generators [ a, b, c ]>
<fp group on the generators [ a, b, c ]>
sage: AbelianInvariants(G);
[ 3, 3, 3 ]
sage: NewmanInfinityCriterion(G,3)
true
Looks wrong? As far as I understand, passing NewmanInfinityCriterion(G,3)
meens that G has arbitrary large homomorphic images in 3-groups. It seems
to contradict with the following (G is the same as above):
sage: G3:=PQuotient(G,3);
sage: h3:=EpimorphismQuotientSystem(G3);
<3-quotient system of 3-class 4 with 9 generators>
[ a, b, c ] -> [ a1, a2, a3 ]
sage: G3:=Image(h3)
<pc group of size 19683 with 9 generators>
I am new user of the GAP. I starting to play with a GAP inside SAGE, 4.4.2,
Lev.
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