[GAP Forum] Quotients of abelian groups
Alexander Hulpke
hulpke at me.com
Mon Mar 8 01:53:36 GMT 2010
Dear Forum,
On Mar 7, 2010, at 7:53 AM, Lyosha Beshenov wrote:
> Given bases of two abelian groups A_1 and A_2, A_2 \subset A_1,
> compute the structure of A_1/A_2.
>
>
> For instance, if A_1 has a basis {a - b, c} and A_2 has a basis
> {a - b - c, -a + b - c}, then A_1/A_2 is isomorphic to Z/2Z.
Assuming you have a basis for A_1, a Smith Normal Form computation will give the desired information.In your example:
gap> m:=[[1,-1,-1],[-1,1,-1]]; #A2generators in terms of A1
[ [ 1, -1, -1 ], [ -1, 1, -1 ] ]
gap> SmithNormalFormIntegerMatTransforms(m);
rec( rank := 2, normal := [ [ 1, 0, 0 ], [ 0, 2, 0 ] ],
rowtrans := [ [ 0, -1 ], [ -1, -1 ] ], rowC := [ [ 1, 0 ], [ 0, 1 ] ],
rowQ := [ [ 0, -1 ], [ -1, -1 ] ],
colC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 1, 1 ] ],
colQ := [ [ 1, -1, 1 ], [ 0, 0, 1 ], [ 0, 1, -1 ] ],
coltrans := [ [ 1, -1, 1 ], [ 0, 0, 1 ], [ 0, 1, 0 ] ] )
normal component shows A_1/A2 =Z/2 x Z
rowtrans and coltrans are the respective base changes.
Best,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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