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Let $G=A*B$. And let $N\unlhd A$ be a normal subgroup of A. Let $H\leq G$ be the kernel of the following map: $$\Psi:A*B\to A/N*1.$$ With Kurosh's Theorem there exists a splitting $H=(H\cap A)*(H\cap B)*F$, where $F\leq G$ is a free subgroup of $G$. Why can I choose $F\neq 1$? Or is $F$ always nontrivial? And if so, why?

Thanks for help.

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    "Splitting" is a bad name for this, since in the context of Group Theory, a "splitting" generally refers to a retract of a projection (that is, expressing the group/subgroup as a semidirect product). "Factorization" is better for this.2012-05-29
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    @Arturo: The term "splitting" is pretty well established now for any graph of groups description of a group $G$, in particular for a free factorization of $G$, or a free factorization with amalgamated subgroup, or an HNN amalgamation, etc.2012-06-02

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