Ensuring commutativity of finite groups (Q2765554)

The following interesting problem is posed and some comments are made on it: Let \(m,n\) be positive integers and \(\mathcal G\) a finite group of order \(g\). Suppose that for all choices of a subset \(M\) of cardinality \(m\) and of a subset \(N\) of cardinality \(n\) in \(\mathcal G\) some member of \(M\) commutes with some member of \(N\) (call this condition Comm). What relations between \(g,m,n\) guarantee that \(\mathcal G\) is Abelian?NEWLINENEWLINENEWLINEHowever, this problem can be posed for all groups, but note that by a famous result of the author (see [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]) one can see that every infinite group satisfying the condition Comm for some positive integers \(m,n\), is Abelian (see \textit{P. Longobardi, M. Maj, A. H. Rhemtulla} [Commun. Algebra 20, No. 1, 127-139 (1992; Zbl 0751.20020)]).NEWLINENEWLINENEWLINEReviewer's remark: The author states that if \(m=1\), \(n=5\) or \(m=2\), \(n=4\), then Comm ensures the group is Abelian, whatever \(g\); which are obviously misprints and the correct values are \(m=1\), \(n=4\) or \(m=2\), \(n=3\). Also the author states that for the groups of order 8 one needs \(m=2\), \(n=5\) to force the group to be Abelian; which is again a misprint and it must be \(m=2\), \(n=4\). And finally the author states a claim that for the groups of order \(p^3\) (\(p\) a prime), the condition Comm with \(m=p\), \(n=p^3-p^2+1\) will ensure commutativity; but the quaternion group of order 8 shows that here is a misprint in the expressions of \(m\) or \(n\) in terms of \(p\).