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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 for some information. – Juan S Feb 28 '12 at 2:18
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 $T$. – Derek Holt Feb 28 '12 at 3:55
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. – David Wheeler Mar 4 '12 at 19:29

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

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