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Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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! For $p=5$, the construction of

Wall, G. E., On Hughes’ $H_{p}$ problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). ZBL0189.31701

shows that we can't beat $\tfrac{1}{25}$ for $p=5$.

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.

Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.

Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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! For $p=5$, the construction of

Wall, G. E., On Hughes’ (H_{p}) problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). ZBL0189.31701

shows that we can't beat $\tfrac{1}{25}$.

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.

Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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! For $p=5$, the construction of

Wall, G. E., On Hughes’ $H_{p}$ problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). ZBL0189.31701

shows that we can't beat $\tfrac{1}{25}$ for $p=5$.

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.

Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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 $N \geq p$. But that's wrong! For $p=5$, the construction of

Wall, G. E., On Hughes’ (H_{p}) problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). ZBL0189.31701

shows that we can't beat $\tfrac{1}{25}$.

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.

Yes, this is the main theorem of

Laffey, Thomas J., The number of solutions of (x^p=1) in a finite group, Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). ZBL0343.20006.

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.

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, Math. Z. 149, 43-45 (1976). ZBL0314.20020.

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! For $p=5$, the construction of

Wall, G. E., On Hughes’ (H_{p}) problem, Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1967). ZBL0189.31701

shows that we can't beat $\tfrac{1}{25}$.

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., J. Algebra 322, No. 3, 791-801 (2009). ZBL1187.20010

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.