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How exactly can I determine if it is normal or not?

MJD
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Jeff
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  • In general, to prove that a subgroup $H$ of a group $G$ is not normal, you need to find $h \in H$ and $g \in G$ such that $ghg^{-1} \not \in H.$ – Geoff Robinson Mar 10 '15 at 14:15
  • Welcome to MSE! Could you add in a line or two telling how you tried to determine its normality? THis will give context as well as an incentive for people to answer it, knowing you're genuine about the question (Of course you are, but people may think otherwise). So it's wise to add a bit of background/context to every question you ask. – Panglossian Oporopolist Mar 10 '15 at 14:17

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Hint:

$$(12)(1234)(12)=(1342)\neq(1234)^k\;,\;\;\text{for any}\;\;k\in\Bbb Z$$

Timbuc
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  • That is too much hint. – MJD Mar 10 '15 at 14:16
  • "Too much"...for whom? I think it is just fine, in particular taking into account that someone making such a basic question must be having problems with this stuff and, thus, he/she is probably a beginner. – Timbuc Mar 10 '15 at 14:17