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Let's say we have a group G containing a normal subgroup H. What are the possible relationships we can have between G, H, and G/H? Looking at groups of small order, it seems to always be the case that G = G/H x H or G/H x| H. What, if any, other constructions/relations are possible? And why is it the case that there are or aren't any other possible constructions/relations(if this question admits a relatively elementary answer)?

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    There are other possibilities. Take $G = \mathbb{Z}/4\mathbb{Z}, H = 2 \mathbb{Z}/4\mathbb{Z}$. Then $G/H = \mathbb{Z}/2\mathbb{Z}$ is not a subgroup of $G$, so $G$ can't be either a direct or semidirect product of $H$ and $G/H$.2011-07-16
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    Anyway, my understanding is that this is actually fairly nontrivial. See http://en.wikipedia.org/wiki/Group_extension#Extension_problem . For finite groups, see http://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem .2011-07-16
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    @Jason: you should go ahead and look at groups of order at least 4: they're much more interesting than groups of "small order".2011-07-16

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