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sorry if this is a basic question, but I was reading about the permutation group $S_3$, and I read that it has three subgroups: $\langle y\rangle$, $\langle xy\rangle$, $\langle x^2y\rangle$ of order 2 and $\langle x\rangle$ of order 3.

Can someone mention which subgroups these are, and how to determine their order? are any of these groups normal?

Angina Seng
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Lana
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    it seems that $y=(12), x=(123)$. Are you familiar with this notation? – 35T41 Apr 16 '17 at 17:34
  • I'm familiar with (12) and (123) – Lana Apr 16 '17 at 17:38
  • Suggestion (not a hint for this particular question). I recommend that you write down all $6$ elements of $S_3$ in cycle notation (which you seem to know how to do) and then do lots of multiplications. For this question: find out what subgroup each of the six elements generates.Then test each for normality, using the definition. – Ethan Bolker Apr 16 '17 at 17:52

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