An Upper Bound for the Probability of Generating a Finite Nilpotent Group
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Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ν(G) =1/n We also classify all groups G with ν(G) = 1/2 Further, we prove that if G is a solvable nonnilpotent group of even order, then ν(G) ≤p+3/4p , where p is the smallest odd prime divisor of |G|, and that equality exists if and only if G/Z∞(G) is isomorphic to the dihedral group of order 2p where Z∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.
Abstract
Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ν(G) =1/n We also classify all groups G with ν(G) = 1/2 Further, we prove that if G is a solvable nonnilpotent group of even order, then ν(G) ≤p+3/4p , where p is the smallest odd prime divisor of |G|, and that equality exists if and only if G/Z∞(G) is isomorphic to the dihedral group of order 2p where Z∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.
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