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I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example:

  • How many normal subgroups does S3 have?
  • How many subgroups of order 4 has group S4?
  • And does group S5 have a subgroup of order 5 or 6?

Is the only way to approach this to try all the possiblities? And how does the calculation differ for normal subgroups?

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  • $\begingroup$ The third part is easy: you can construct elements of $S_5$ that have orders 5 and 6. $\endgroup$
    – Jessica B
    Jan 28, 2015 at 7:59
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    $\begingroup$ In general, finding the number of subgroups (normal or otherwise) is really hard (but fortunately these are small examples, where one can easily just try by hand). $\endgroup$
    – Tobias Kildetoft
    Jan 28, 2015 at 8:00

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