On a bound of Cocke and Venkataraman

Abstract

<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 >\n <mml:mrow>\n <mml:mo>gcd</mml:mo>\n <mml:mo>(</mml:mo>\n <mml:mi>m</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </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 >\n <mml:mrow>\n <mml:mrow>\n <mml:mo>|</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo>|</mml:mo>\n </mml:mrow>\n <mml:mo>≤</mml:mo>\n <mml:mi>q</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>m</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msup>\n <mml:mi>k</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo>/</mml:mo>\n <mml:mi>φ</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>m</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:mrow>\n </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 >\n <mml:mi>φ</mml:mi>\n </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 >\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo><</mml:mo>\n <mml:mn>36</mml:mn>\n </mml:mrow>\n </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>