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Topic: Let G be a group , N a normal subgroup of G
Replies: 13   Last Post: Feb 6, 2013 6:11 AM

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 quasi Posts: 11,770 Registered: 7/15/05
Re: Let G be a group , N a normal subgroup of G
Posted: Feb 5, 2013 4:00 PM

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