quasi wrote: >Dan wrote: > >>Does there always exist a subgroup H of G such that G = NH, and >>(H intersection N) = the identity element? > >Presumably you intended to require that N be nontrivial and >proper. > >But even with that restriction, the answer is still no. > >For example, let G = Z (the additive group of integers). Since >Z is abelian, all subgroups of Z are normal. Let N be any >nontrivial proper subgroup. Suppose H is a subgroup of Z such >that > > G = H + N > > (H intersect N) = 0 > >Since G = H + N, it follows that N is nontrivial.
Since G = H + N, it follows that H is nontrivial.
>Since N,H are both nontrivial subgroups of Z, it follows that >(H intersect N) is nontrivial, contradiction.