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Consider a quotient group $G/H$. If there is a section that is a subgroup of $G$ (I mean a transversal that is also a group), must the group be necessarily a semidirect (including direct) product?

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    note TH =$G$and $T\cap H = \{e\}$ and$H$is normal. one can define the homomorphism from$T$to Aut(H) explicitly by conjugating h in$H$by t in T.2012-03-04

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As the enterprise of offering redemption to tormented questions out of unanswered questions's hell requires, I'll just write down as an answer the comments by Derek and David:

We have a group $\,G\,\,,\,\,H\triangleleft G\,$ and $\,T\,$ a transversal of $\,H\,$ in $\,G\,$, meaning: $G/H=\{tH\;\;;\;\;t\in T\}$

If $\,T\leq G\,$, then$G=TH\,\,,\,\,T\cap H=\{1\}\,\,and\,\,H\triangleleft G\Longrightarrow G=T\rtimes H$