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Say h:G-->G' is a homomorphism between groups, and that we know a set

of generators {ki} for Imh:=h(G)<G' , and we also know of a set of generators

{b_j} for Kerh . Can we use these two sets {ki} and {bj} of generators for

Imh and Kerh respectively, to produce a set of generators for G itself?

It looks a bit like the group extension problem (which I know very little about,

unfortunately).

This is what I have tried so far :

We get a Short Exact Sequence:

1 -->Kerh -->G-->Imh -->1

But I am not sure this sequence necessarily splits (if it doesn't split, then you must acquit!)

It would seem like we could pull-back generators of Imh back into G, i.e., for any g in G, we can write h(g)=Product{$k_i$ $e_i$} of generators in h(G).

Similarly, we know that G/Kerh is Isomorphic to h(G) , and that g~g' iff h(g)=h(g') ( so that,the isomorphism h':G/Kerh-->h(G) is given by h'([g]):=h(g) )

But I get kind of lost around here.

Any Ideas?

Thanks.

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# Finding a set of Generators for a group G when Generators for Kerh, Imh are known; h

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