Groups with relatively large commutativity measure (Q1390415)

Let \(G\) be a finite group, \(k(G)\) the class number of \(G\). The number \(\text{mc}(G)=k(G)/| G|\) is said to be the commutativity measure of \(G\). It is known that whenever \(H\leq G\), then \(\text{mc}(H)\geq\text{mc}(G)\) (i.e., \(\text{mc}(*)\) is monotone), and if \(N\) is normal in \(G\) then \(\text{mc}(G/N)\geq\text{mc}(G)\). Next, \(\text{mc}(G)=1\) if and only if \(G\) is abelian. In this note, the groups \(G\) satisfying \(\text{mc}(G)\geq\tfrac 1p\), where \(p\) divides \(| G|\), are studied. The aim of this note is to prove the following nontrivial Main Theorem. Let \(p\) be a prime divisor of the order of a group \(G\) such that a Sylow \(p\)-subgroup of \(G\) is not contained in \(Z(G)\), the center of \(G\). If \(\text{mc}(G)\geq\tfrac 1p\), then \(G\) is \(p\)-solvable and \(| P'|\leq p\), where \(P\) is a Sylow \(p\)-subgroup of \(G\). If \(r>p\) is a prime divisor of \(| G|\), the Sylow \(r\)-subgroup of \(G\) is abelian and normal. Furthermore, one of the following assertions holds: (a) \(P\) is normal in \(G\). (b) \(G=A\times H\), where \(H\) is abelian, \(A=P_0\cdot A'\) is minimal nonabelian, \(P_0\leq P\) and either \(p=| A'|-1\) or \(p=2\) and \(| A'|=3\). The author produces the proof independent of character theory. In the proof of \(p\)-solvability of \(G\) one uses Burnside's theorem on doubly transitive groups of prime degree. Using properties of minimal nonnilpotent groups and the fact that the function \(\text{mc}(*)\) is monotone, one can shorten some proofs (for example, proofs of Lemmas 1.2 and 1.4). For related results see \textit{Y. G. Berkovich} and \textit{E. M. Zhmud} [Characters of finite groups. Parts 1, 2 (Transl. Math. Monogr. 172, 181), Am. Math. Soc., Providence (1998), Chapters 11 and 28].