[GAP Forum] Quotients of abelian groups

[Alexander Hulpke]

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,
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