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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: 12,067 Registered: 7/15/05
Re: Let G be a group , N a normal subgroup of G
Posted: Feb 5, 2013 11:52 AM

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