Order of cyclic subgroups in symmetric groups

Question

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 $(1\ 2\ 3\ 4\ 5\ 6)$?

Might it be that you think that "maximum element order" and "largest size of a cyclic subgroup" are two different things? — the_fox, Apr 29 '18 at 21:02

score 2 · Answer 1 · answered Apr 29 '18 at 20:57

2

Hint:

The order of a cyclic subgroup is the order of any of its generators.
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 $(1\ 2)(3\ 4\ 5\ 6\ 7)$.

answered Apr 29 '18 at 20:57

Arnaud Mortier

27,276

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

Order of cyclic subgroups in symmetric groups

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