<jats:p>Let <jats:italic>G</jats:italic> be a finite group with exactly <jats:italic>k</jats:italic> elements of largest possible order <jats:italic>m</jats:italic>. Let <jats:italic>q</jats:italic>(<jats:italic>m</jats:italic>) be the product of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\gcd (m,4)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>gcd</mml:mo>
<mml:mo>(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>,</mml:mo>
<mml:mn>4</mml:mn>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> and the odd prime divisors of <jats:italic>m</jats:italic>. We show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$|G|\le q(m)k^2/\varphi (m)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mrow>
<mml:mo>|</mml:mo>
<mml:mi>G</mml:mi>
<mml:mo>|</mml:mo>
</mml:mrow>
<mml:mo>≤</mml:mo>
<mml:mi>q</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:msup>
<mml:mi>k</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>/</mml:mo>
<mml:mi>φ</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\varphi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>φ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> denotes Euler’s totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with <jats:inline-formula><jats:alternatives><jats:tex-math>$$k<36$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo><</mml:mo>
<mml:mn>36</mml:mn>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>. This is motivated by a conjecture of Thompson and unifies several partial results in the literature.</jats:p>
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