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

by Bacle
Tags: generators, kerh
 P: 662 Hi, Algebraists: Say h:G-->G' is a homomorphism between groups, and that we know a set of generators {ki} for Imh:=h(G)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.
 Mentor P: 18,293 My thoughts are this: Take $$\{a_1,...,a_n\}$$ generators of ker(f), and take $$\{b_1,...,b_m\}$$generators of im(f). For every bi, we can find a ci such that $$f(c_i)=b_i$$. Then $$\{a_1,...,a_n,b_1,...,b_m\}$$ is a generating set for G. Indeed, take g in G, then we can write f(g) as $$f(g)=b_{i_1}...b_{i_j}=f(c_{i_1})...f(c_{i_j})=f(c_{i_1}...c_{i_j})$$ Thus $$f(gc_{i_1}^{-1}...c_{i_j}^{-1})\in Ker(f)$$, so we can write $$gc_{i_1}^{-1}...c_{i_j}^{-1}=a_{k_1}...a_{k_l}$$, so that follows $$g=c_{i_1}...c_{i_j}a_{k_1}...a_{k_l}$$ We have writte g as a combination of the suitable elements, so the set $$\{a_1,...,a_n,c_1,...,c_m\}$$ is generating...