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Is $\{(1),(1,3)\}$ a normal subgroup in $S_3$? I know that a normal subgroup means that the left cosets are equal to the right cosets.

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    $\begingroup$ You should test this by hand; there aren't too many multiplications to perform. $\endgroup$ – apnorton May 14 '15 at 2:47
  • $\begingroup$ @Gamamal answer is correct, and it uses your definition of 'normality' in a n equivalen (but slightly different) way. You say that $H\leq G$ is normal if, and only if, for all $x\in G$ we have $xH=Hx$. With only few lines of proof you can show that $H$ is normal in $G$ if, and only if, $x^{-1}Hx=H$. $\endgroup$ – rafforaffo May 14 '15 at 13:20
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$(12)^{-1}(13)(12)=(12)(13)(12)=(23)$ which is not in the subgroup. Hence the subgroup is not normal.

The subgroup $\{e,(123),(132)\}$ is normal however. You should test it by hand. I did not do this however because there is a theorem that tells you if a subgroup has exactly half of the elements then it is normal.

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