# Order of cyclic subgroups in symmetric groups [duplicate]

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

• It's not asking the same thing – Hopper Apr 29 '18 at 20:59
• In a way, yes, it is. – Arnaud Mortier Apr 29 '18 at 21:00
• I guess I'm just not understanding then – Hopper Apr 29 '18 at 21:01
• 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
• Possibly, yeah. – Hopper Apr 29 '18 at 21:10

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 $(1\ 2)(3\ 4\ 5\ 6\ 7)$.
• @Hopper Yes!${}$ – Arnaud Mortier Apr 29 '18 at 21:13