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What is the largest possible order of a cyclic subgroup of $S_7$?

What's an example of this?

I really just need a better understanding of cyclic subgroups of symmetric groups. I know that the largest possible subgroup of $S_7$ is of size $7$. And I know all the possible cycle structures. Would it be $6$ because $\left<(1 2 3 4 5 6 )\right>$ is a cyclic subgroup generated by the size of $S_6$ or the element $(1\ 2\ 3\ 4\ 5\ 6)$?

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  • $\begingroup$ It's not asking the same thing $\endgroup$ – Hopper Apr 29 '18 at 20:59
  • $\begingroup$ In a way, yes, it is. $\endgroup$ – Arnaud Mortier Apr 29 '18 at 21:00
  • $\begingroup$ I guess I'm just not understanding then $\endgroup$ – Hopper Apr 29 '18 at 21:01
  • $\begingroup$ Might it be that you think that "maximum element order" and "largest size of a cyclic subgroup" are two different things? $\endgroup$ – the_fox Apr 29 '18 at 21:02
  • $\begingroup$ Possibly, yeah. $\endgroup$ – Hopper Apr 29 '18 at 21:10
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Hint:

  1. The order of a cyclic subgroup is the order of any of its generators.
  2. The order of an element of $S_n$ is the lcm of the lengths of the cycles in its disjoint cycle decomposition.

Example: in $S_7$ you have a cyclic subgroup of order $10=2\times 5$ generated by $(1\ 2)(3\ 4\ 5\ 6\ 7)$.

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  • $\begingroup$ Does this mean it could be order 12? Because 4 × 3 is generated by (1234)(567) $\endgroup$ – Hopper Apr 29 '18 at 21:03
  • $\begingroup$ @Hopper Yes!${}$ $\endgroup$ – Arnaud Mortier Apr 29 '18 at 21:13
  • $\begingroup$ Thank you then! $\endgroup$ – Hopper Apr 29 '18 at 21:14

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