Date: Feb 5, 2013 11:52 AM
Author: quasi
Subject: Re: Let G be a group , N a normal subgroup of G

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 N,H are both nontrivial subgroups of Z, it follows that
(H intersect N) is nontrivial, contradiction.

quasi