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quasi
Posts:
11,254
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



