Suppose $n\_p(G)$ is the number of elements of order $p$ in a group $G$. 


Does there for every prime $p$ exist a $\epsilon\_p > 0$, such that for any group $G$ $n\_p(G) > (1 - \epsilon\_p(G))|G|$ implies, that $G$ is a $p$-group?


This question was inspired by a well known fact, that for any group $G$ $n\_2(G) > \frac{3}{4}|G|$ implies $G$ being a $2$-group (or, alternatively, that $\epsilon\_2 = \frac{1}{4}$).



A question on "$p$-group bounds"?
Yes, this is the main theorem of 


*Laffey, Thomas J.*, [**The number of solutions of (x^p=1) in a finite group**](http://dx.doi.org/10.1017/S0305004100052865), Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). [ZBL0343.20006](https://zbmath.org/?q=an:0343.20006).


Namely, if a finite group $G$ has more than $\tfrac{p}{p+1} |G|$ elements of order $p$, then $G$ is a $p$-group. This bound is achieved by $C\_p \ltimes C\_2^k$ where $p = 2^k-1$ is a Mersenne prime.




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There is a related very frustrating problem: Is there an $\delta\_p$ such that, if $G$ has more than $1-\delta\_p$ elements of order $p$, then $G$ is $p$-torsion? We can take $\delta\_3 = \tfrac{2}{9}$, see 


*Laffey, Thomas*, [**The number of solutions of (x^3=1) in a 3-group**](http://dx.doi.org/10.1007/BF01301629), Math. Z. 149, 43-45 (1976). [ZBL0314.20020](https://zbmath.org/?q=an:0314.20020).


I thought for a bit that the right bound might be $\tfrac{p-1}{p^2}$ in general, which occurs for $C\_p \ltimes \mathbb{Z}[\zeta\_p]/(1-\zeta\_p)^N$ where $\zeta\_p$ is a $p$-th root of unity and $p \leq N \leq 2p-2$. But that's wrong! Wall constructed a $5$-group of exponent $25$ where all the elements of order $25$ were in an index $25$ subgroup, demonstrating that $\delta\_5$, if it exists, is $< \tfrac{1}{25}$.


*Wall, G. E.*, On Hughes’ $H\_{p}$ problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). [ZBL0189.31701](https://zbmath.org/?q=an:0189.31701)


But, as far as I know, no one knows whether any such $\delta\_5$ exists at all! 


See 
*Havas, George; Vaughan-Lee, Michael*, [**On counterexamples to the Hughes conjecture.**](http://dx.doi.org/10.1016/j.jalgebra.2009.04.011), J. Algebra 322, No. 3, 791-801 (2009). [ZBL1187.20010](https://zbmath.org/?q=an:1187.20010)


for groups achieving $\tfrac{1}{p^2}$ for various $p>5$. I will also note that I talked to Harry Altman about a lot of this material.


