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

I meant:

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.

quasi