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If we write a partition $n=k_1+...+k_r$, then we can create a $(k_1,...,k_r)$-cycle in $S_n$ with order equal to the least common multiple of the $k_i$'s. It is clear that every cyclic subgroup will arise this way, by considering the cycle type of a generator. So this appears to give a classification of which cyclic subgroups can occur in $S_n$, namely $Z/mZ \hookrightarrow S_n$ iff $m$ is the lcm of numbers whose sum is $\leq n$. I'd like a (computationally at least) cleaner criterion. So...

$\bullet$ What are the (orders of) maximal cyclic groups occurring in $S_n$ (under containment)? In particular, what is the largest-order cyclic subgroup in $S_n$?

In light of the above, the latter question amounts to asking for a formula for the maximum value (over all partitions) of the function from partitions of $n$ to $\mathbb{N}$ which outputs the lcm of the terms in the partition. The former question asks for local maximum values (under divisibility) of this function. Statements about the partitions giving rise to these values would also be of interest.

In a slightly different direction, I also wondered about the following:

$\bullet$ How many different (up to iso) cyclic subgroups are in $S_n$? This is deducible from an answer to Q1 in terms of the number-of-divisors function, but I'd be interested in arguments that go a different route.

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What do you mean by "up to isomorphism"? Do you mean up to isomorphism as abstract cyclic groups? – Qiaochu Yuan Nov 11 '12 at 22:22
Yes, up to abstract isomorphism. – Jon Cohen Nov 11 '12 at 22:30

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See also – Gerry Myerson Nov 11 '12 at 22:31

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