Date: Feb 5, 2013 4:00 PM
Subject: Re: Let G be a group , N a normal subgroup of G
>>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
>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
> 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.