Abstract
Let m, n be positive integers and ๐ be a class of groups. We say that a group G satisfies the condition ๐(m, n), if for every two subsets M and N of cardinalities m and n, respectively, there exist x โ M and y โ N such that โจx, yโฉ โ ๐. In this article, we study groups G satisfies the condition ๐(m, n), where ๐ is the class of nilpotent groups. We conjecture that every infinite ๐(m, n)-group is weakly nilpotent (i.e., every two generated subgroup of G is nilpotent). We prove that if G is a finite non-soluble group satisfies the condition ๐(m, n), then , for some constant c (in fact c โค max{m, n}). We give a sufficient condition for solubility, by proving that a ๐(m, n)-group is a soluble group whenever m + n < 59. We also prove the bound 59 cannot be improved and indeed the equality for a non-soluble group G holds if and only if G โ
A
5, the alternating group of degree 5.
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ACKNOWLEDGMENTS
I would like to thank the referees for their useful comments. This research was in part supported by a grant from IPM (No. 89200028).
Notes
Communicated by A. Olshanskii.