[This problem](https://math.stackexchange.com/questions/1278465/maximum-number-of-sylow-subgroups/) (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the problem came from, even though he/she has been asked. Has anyone seen this result before? If so, where?



Origin of group theory problem (bound on number of Sylow subgroups)
Since this question has resurfaced (after 3 years) let me
say something about the origin of this Monthly problem.




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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 $20$ --- $A\_5$*.




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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.




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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.














