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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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    I'm not exactly sure what the question is but I'm guessing the answer has something to do with the splitting lemma. See [here](http://mathworld.wolfram.com/GroupExtension.html) for some information.2012-02-28
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    If there is a transversal $T$ of $H$ in $G$ that is a subgroup of $G$, then $T$ is a complement of $H$ in $G$, so the extension is split, and is isomorphic/equal to a semidirect product of $H$ by $T$.2012-02-28
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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$$G=TH\,\,,\,\,T\cap H=\{1\}\,\,and\,\,H\triangleleft G\Longrightarrow G=T\rtimes H$$