On a property of cosets in a finite group (Q1101824)

Let G be a finite group, \(H\triangleleft G\) and \(g\in G\). Let \(b\in gH\) be such that \(| b| =\min \{| x|:\) \(x\in gH\}\). The author is concerned with the following question: under what conditions \(| b| | | x|\) for every \(x\in gH?\) This is shown to be true if (Th. 1) H is nilpotent, or (Th. 2) if gH is a p-element of G/H. There is a misprint which does not affect the proof of Theorem 1: instead of \(h_ i'=g_ i\prod^{n-1}_{k=0}h_ i^{g^ k}\) read \(h_ i'=g_ i\prod^{n-1}_{k=0}h_ i^{g^{n-1-k}}\).