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I am quite new here, as my number of posts might indicate. Thus I am not really sure whether or not this question should be posted here or somewhere else. It is not a homework, but neither is it a question that could not be a homework. However, here we go.

I have, during a course in Abstract algebra, solved problems of the type:

Compute, i.e. classify according to the Decomposition Theorem (DT) for finitely generated abelian groups, The factor group (Z*Z*Z) / <(3,3,3)>.

The solution to this problem is not very hard. One just constructs a homomorphism f: Z*Z*Z --> Z*Z*Z having kernel <(3,3,3)>, and then uses the theorem (sometimes called the fundamental homomorphism theorem, sometimes the first isomorphism theorem) that states that if f:G --> G' is a homomorphism with ker(f) = H, Then G/H is isomorphic to Im(f). In the case above, Im(f) would be (Z/3Z)*Z*Z, or some isomorphic group that is later translated to the appropriate form.

The normal subgroup <(3,3,3)> can be alternated to make this problem more cumbersome.

One case I have not yet managed to complete is when <(3,3,3)> is replaced by H = <(6,4,8)>. Here I seem to fail when trying to restrict my homomorphism to some subset of the original codomain.

Now, if the problem had been to compute for example (Z*Z*Z*Z*Z*Z) / <(6,4,8,5,12)> I would certainly give up before even trying. Constructing a homomorphism with the appropriate kernel is perhaps not too hard, but finding the image... I would be not too cheerful at the thought.

Let's, for simplicity of notation, let G = Z*Z*...*Z, H = <(a1, a2, ..., an)> and Let K be the group isomorphic to G/H expressed as a product of cyclic groups according to DT.

My question is, has anyone got any ideas on how to simplify the work here? Could conclude something about the Betti number of K just be regarding the dimension of G and the integers a_i, (i = 1,2,...,n)? If the ai are all equal, or such that they are all multiples of one a_j, then one can just construct easily homomorphism reducing the a_j'th coordinate of G (mod aj), and then manipulate the other coordinated by some linear combination such that the appropriate kernel is obtained, and it is easy to find a proper restriction of the homomorphism, so that one can apply the fundamental homomorphism property.

I would appreciate hints and tips from more knowledgable people, as I would like to gain some understanding of this type of quotient groups, and it would simplify computations.

Thanks in advance

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# Problem regarding computation of factor groups.

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