TITLE: Order of products of elements in symmetric groups
QUESTION [30 upvotes]: Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order $c$?
The assertion is true at least for $n \leq 10$, see
here.
Update on Sep 2, 2015: On Aug 10, 2015
Joachim König has posted
a preprint to the arXiv which gives
a positive answer to the question. Assuming that this preprint is correct,
this completely answers the question -- and thus also Problem 18.49 in the
Kourovka Notebook.
Update on Jun 18, 2014: The assertion is true at least for $n \leq 50$, see
here (4MB text file).
The list of examples in GAP-readable format can be found
here.
Added on Dec 11, 2013: This question will appear as
Problem 18.49 in:
Kourovka Notebook: Unsolved Problems in Group Theory. Editors V. D.
Mazurov, E. I. Khukhro. 18th Edition, Novosibirsk 2014.
Added on Nov 24, 2013: Is there really not enough known about, say, the
class multiplication coefficients of ${\rm S}_n$ to answer this question?
Text of the question as of Feb 12, 2013:
This question is a follow-up on Order of elements  .
Derek Holt's answer to that question is nice, but it seems that the degree of the permutations it gives is a lot larger than necessary.
So, given natural numbers $m, n, k > 1$, what is the smallest $d$ such that the symmetric group of degree $d$ has elements of order $m$ and $n$ whose product has order $k$? -
Clearly if the largest of the numbers $m$, $n$, $k$ is prime, then $d$ must be at least
$\max(m,n,k)$, and there are some cases where $d$ actually must be larger. However a quick computation suggests that $d = \max(m,n,k) + 2$ might work always. - But does this or a similar bound hold?
EDIT: Smallest-degree examples for all $m, n, k \leq 8, m \leq n$ can be found
here.

REPLY [9 votes]: The main theorem in a paper of G. A. Miller [1] is the following:

THEOREM. If $l, m, n$ are any three integers greater than unity, of which we
call the greatest $k$, it is always possible to find three substitutions $(L, M, N)$ of $k + 2$ or some smaller number of elements and of orders $l, m, n$ respectively such that $LM=N$.

Which gives a positive answer to your question.
Papers [3], [4], [5] by Brenner and Lyndon give a different proof of Miller's result, and consider the problem of finding for $l,m,n > 1$ the smallest $d$ such that $S_d$ contains permutations $x,y$ with $(|x|, |y|, |xy|) = (l,m,n)$. Other related results are e.g. in [2] and [6].
Papers [8], [9] also give a proof of the theorem by Miller. Also in [7], although it seems without assuming permutations of degree $\leq \max(l,m,n)+2$ in some cases.
The main theorem of [10] constructs for $l,m,n > 1$ elements $A,B \in PSL(2,q)$ (for suitable $q$) such that $|A| = l$, $|B| = m$, and $|AB| = n$. The construction pointed out by Derek Holt here is similar but a bit shorter. Yet another construction with matrices is in [11].
References:
[1] MR1505829 G. A. Miller, On the Product of Two Substitutions. Amer. J. Math. 22 (1900), no. 2, 185–190. JSTOR
[2] MR1505882 G. A. Miller, Groups Defined by the Orders of Two Generators and the Order of their Product. Amer. J. Math. 24 (1902), no. 1, 96–100. JSTOR
[3] MR0767585 J. L. Brenner, R. C. Lyndon, A theorem of G. A. Miller on the order of the product of two permutations. I. Jñānābha 14 (1984), 1–16. link
[4] MR0809272 J. L. Brenner, R. C. Lyndon, A theorem of G. A. Miller on the order of the product of two permutations. II. The minimal degree in case a=2. Indian J. Math. 26 (1984), no. 1-3, 105–133 (1985).
[5] MR0748120 J. L. Brenner, R. C. Lyndon, A theorem of G. A. Miller on the order of the product of two permutations. III. The minimal degree in case a>2. Pure Appl. Math. Sci. 20 (1984), no. 1-2, 37–51.
[6] MR0743150 J. L. Brenner, R. C. Lyndon, The orbits of the product of two permutations. European J. Combin. 4 (1983), no. 4, 279–293. DOI
[7] MR0053937 R. H. Fox, On Fenchel's conjecture about F-groups. Mat. Tidsskr. B 1952 (1952), 61–65. JSTOR
[8] MR3508006 J. König, A note on the product of two permutations of prescribed orders. European J. Combin. 57 (2016), 50–56. DOI
[9] MR3694453 J. Pan, On a conjecture about orders of products of elements in the symmetric group. J. Pure Appl. Algebra 222 (2018), no. 2, 291–296. DOI
[10] MR0283093 R. D. Feuer, Torsion-free subgroups of triangle groups. Proc. Amer. Math. Soc. 30 (1971), 235–240. DOI
[11] MR0207852 J. Mennicke, Eine Bemerkung über Fuchssche Gruppen. Invent. Math. 2 (1967), 301–305. DOI