Motivated by a classic result for free groups, one says that a group G has the Magnus property if the following holds: Whenever two elements generate the same normal subgroup of G, they are conjugate or inverse-conjugate in G. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most 2. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free center-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most 2. On the way, we display 2-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property.
Background There is a high prevalence of shoulder injuries in volleyball and their preventive management is still a challenge. Isokinetic assessment is largely used in sports injury prevention. Few studies investigate the correlation between the isokinetic profi le and injuries story, furthermore these studies analyses the agonist/antagonist ratio and do not consider other parameters as fatigue.Objective to investigate the differences between injured and non injured dominant shoulder in the lateral/medial rotators ratio and fatigue index in volleyball male athletes. Setting all testing took place in the Sports Injury Prevention and Rehabilitation Laboratory-LAPREV in elite volleyball. Participants 42 athletes (mean+SD age, 21.3+4.1; height, 196.8+0.06 and body mass, 89.9+8.8) were evaluated in this study (28 at injured group). Participants with upper-extremity surgery in the previous 6 months and pain during the isokinetic test were excluded. Assessment of risk factor: agonist and antagonist ratio at 60°/s and 360°/s and index fatigue at 360°/s in 90° of abduction at dominant shoulder. Main outcome measurement trauma and overuse shoulder injuries story provided by questionnaire. ResultsThe results showed statistical differences between groups in lateral rotators (p=0.039) and medial rotators fatigue (p=0.024). The mean fatigue of lateral rotators in the injury group was 52.9 and 37.0 for non-injured group and 31.6 and 43.8 respectively for medial rotators. No differences were found in the ratio analysis between groups (p=0.403 for 60°/s and p=0.289 for 360°/s). The means were 71.6 and 67.4 for injured and non-injured groups at 60°/s and 56.9 and 63.9 respectively at 360°/s. Conclusion The results show that lateral rotators fatigue was greater in the injured group and no differences were found in the ratio parameters. The fatigue analysis must be incorporated in the athlete´s assessment, because seems to be more informative about injury risk.
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