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Let $G, H$ be groups and $A\subset G, B\subset H$ be subgroups. And let $\phi:B\to Aut(A)$ be any homomorphism. Define a group $K=G*H/N$ where $N$ is the normal subgroup in the free product $G*H$ generated by $\lbrace b^{-1}ab=\phi(b)(a) | a\in A, b\in B \rbrace$.

Then this is a generalization of both semidirect product when $A=G, B=H$, and HNN extension when $B=H=\langle t\rangle$.

Is there any reference about this kind of product? For example, what the normal form for given word is, when this group has a torsion, when two given elements are conjugate, or commute.

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(Sorry, I can't make the symbols work.) Isn't this just $G*_A A\rtimes B*_B H$? –  nolte Sep 19 '11 at 11:56
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