Bounding the number of classes of a finite group in terms of a prime

Héthelyi and Külshammer showed that the number of conjugacy classes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {k(G)} of any solvable finite group G whose order is divisible by the square of a prime p is at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mn>49</m:mn> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mn>60</m:mn> </m:mrow> </m:math> {(49p+1)/60} . Here an asymptotic generalization of this result is established. It is proved that there exists a constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {c>0} such that, for any finite group G whose order is divisible by the square of a prime p , we have <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> {k(G)≥ cp} .