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](https://stefan-kohl.github.io/problems/permutation_product_examples.txt).


**Update on Sep 2, 2015:** On Aug 10, 2015
[Joachim König](https://mathsci.kaist.ac.kr/%7Ejkoenig/) has posted
a [preprint](https://arxiv.org/abs/1508.02170) 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](https://arxiv.org/abs/1401.0300).


**Update on Jun 18, 2014:** The assertion is true at least for $n \leq 50$, see
[here](https://stefan-kohl.github.io/problems/permutation_product_examples_50.txt) (4MB text file).


The list of examples in GAP-readable format can be found
[here](https://stefan-kohl.github.io/problems/permutation_product_examples_50.g).


**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](https://mathoverflow.net/questions/118035/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](https://stefan-kohl.github.io/problems/mnk_examples.txt).



Order of products of elements in symmetric groups
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](https://mathoverflow.net/questions/118035/order-of-elements) is similar but a bit shorter. Yet another construction with matrices is in [11].


References:


[1] [MR1505829](https://mathscinet.ams.org/mathscinet-getitem?mr=1505829) G. A. Miller, On the Product of Two Substitutions. Amer. J. Math. 22 (1900), no. 2, 185–190. [JSTOR](https://www.jstor.org/stable/2369754)


[2] [MR1505882](https://mathscinet.ams.org/mathscinet-getitem?mr=1505882) 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](https://www.jstor.org/stable/2370009)


[3] [MR0767585](https://mathscinet.ams.org/mathscinet-getitem?mr=767585) 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](http://docs.vijnanaparishadofindia.org/jnanabha/jnanabha_volume_14_1984/1.pdf)


[4] [MR0809272](https://mathscinet.ams.org/mathscinet-getitem?mr=809272) 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](https://mathscinet.ams.org/mathscinet-getitem?mr=748120) 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](https://mathscinet.ams.org/mathscinet-getitem?mr=743150) J. L. Brenner, R. C. Lyndon, The orbits of the product of two permutations. European J. Combin. 4 (1983), no. 4, 279–293. [DOI](https://doi.org/10.1016/S0195-6698(83)80024-9)


[7] [MR0053937](https://mathscinet.ams.org/mathscinet-getitem?mr=53937) R. H. Fox, On Fenchel's conjecture about F-groups. Mat. Tidsskr. B 1952 (1952), 61–65. [JSTOR](https://www.jstor.org/stable/24530069)


[8] [MR3508006](https://mathscinet.ams.org/mathscinet-getitem?mr=3508006) J. König, A note on the product of two permutations of prescribed orders. European J. Combin. 57 (2016), 50–56. [DOI](https://doi.org/10.1016/j.ejc.2016.03.006)


[9] [MR3694453](https://mathscinet.ams.org/mathscinet-getitem?mr=3694453) 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](https://doi.org/10.1016/j.jpaa.2017.04.001)


[10] [MR0283093](https://mathscinet.ams.org/mathscinet-getitem?mr=283093) R. D. Feuer, Torsion-free subgroups of triangle groups. Proc. Amer. Math. Soc. 30 (1971), 235–240. [DOI](https://doi.org/10.1090/S0002-9939-1971-0283093-6)


[11] [MR0207852](https://mathscinet.ams.org/mathscinet-getitem?mr=207852) J. Mennicke, Eine Bemerkung über Fuchssche Gruppen. Invent. Math. 2 (1967), 301–305. [DOI](https://doi.org/10.1007/BF01425406)


