Origin of group theory problem (bound on number of Sylow subgroups)
The problem seems to have appeared as Problem 11856 in American Mathematical Monthly in the July 2015 Issue, proposed by K. Kearnes (USA).
Since this question has resurfaced (after 3 years) let me say something about the origin of this Monthly problem.
A few years back a colleague gave me the task of making
up the algebra preliminary exam for our first-year
graduate students. Among the group theory problems that made
my initial list were:
(1) Show that there is no finite group with more Sylow subgroups than elements.
(This later evolved to: Let $G$ be a finite group. Show that the number of nontrivial Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$.)
(2) Call a positive integer $c$ curious if there is a nontrivial finite group $G$ such that, for every prime $p$ dividing $|G|$, the number of Sylow $p$-subgroups is exactly $c$. Find all curious integers.
(3) You are told that, for certain $n$ and $p$, $S_n$ has exactly $n$ Sylow $p$-subgroups. What are the possible pairs $(n, p)$?
(4) Prove or disprove: There is no group $G$ of square-free order such that, for every prime $p$ dividing $|G|$, the number of Sylow $p$-subgroups is $p + 1$.
(5) Let $G$ be a finite group and let $S\subseteq G$ be a subset. Show that if $S$ has nonempty intersection with each conjugacy class of $G$, then the subgroup generated by $S$ is $G$.
(6) Show that if the conjugacy classes of $G$ have size at most $4$, then $G$ is solvable.
I ultimately used Problem (6) on the exam. My notes to myself say about (6): Here $4$ can be replaced by any number up to $19$, but the statement becomes false for $20$ --- $A_5$.
I decided Problem (1) was too hard for first-year grad students, so I sent it to the problems section of the Monthly. My solution to Problem (1) was similar to the one by Richard Stong, which was published by the Monthly. In particular, I also used Burnside's normal complement theorem and the estimate $\sum_{p}\frac{1}{p^{2}} < \frac{1}{2}$.
Once the problem was type-set by the Monthly, I got a chance to proofread it. I sent my final OK to the Monthly on May 10, 2015. Strangely, the problem appeared on the Art of Problem Solving website the next day, worded in exactly the same way. The day after that it appeared at MSE, worded exactly the same way. It did not appear in print in the Monthly for another 3 months.
One last comment about this problem. When formulating the problem
I asked myself: is it possible to show that there is an injective
function $f:\textrm{Syl}(G)\to G$ such that $f(P)\in P$ for all $P$?
I don't know the answer to this version of the question.
This is not particularly helpful, but in the paper by Pyber, Asymptotic results for finite simple groups on page 309 of this conference proceedings, the author states on page 320:
J.P. Zhang proved that the total number of Sylow subgroups of $G$ is at most $|G|-1$ (personal communication).
Strictly speaking the number of Sylow subgroups of $S_3$ is $5$ rather than $4$, because the trivial subgroup is a Sylow $p$-subgroup for all primes other than $2$ or $3$, so this bound is in that sense the best possible. But of course it is less good than the $2/3$ bound, and $S_3$ is the only example where this is achieved.