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We say $ G$ is the Zappa-Szep product of two subgroups $ K$ and $ P$ if $ K\cap P = \{e\}$ and the function $ K\times P \to G$ , $ (k,p)\mapsto kp$ , is bijective. The Iwasawa decomposition shows that we can have amenable $ K$ and $ P$ such that $ G=KP$Read more