Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime . We say that is -supplemented in if there is a subgroup of such that = and ∩ ≤ , where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in . In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in ( ).
Background: Patients after total hip arthroplasty (THA) often suffered moderate or even severe pain, seriously affecting the early postoperative recovery. This study aimed to investigate the analgesic efficacy of ultrasound-guided transmuscular quadratus lumborum block (T-QLB) combined with fascia iliaca compartment block (FICB) for elderly patients undergoing THA.Methods: Sixty-four patients scheduled for THA were included in this randomized controlled study. The patients were divided into two groups: group Q and group QF. Before anesthesia induction, group Q was injected with 0.375% ropivacaine 40ml. In the QF group, T-QLB combined with FICB was injected with 0.375% ropivacaine 20ml, respectively. Paracetamol 1g regularly at 6 h intervals and patient-controlled intravenous analgesia (PCIA) were administrated in both groups postoperatively. The primary outcome was cumulative sufentanil consumption via PCIA on postoperative 24 h. The secondary outcomes included pain degree, time to the first analgesic requirement, range of motion, quality of recovery, and the incidence of postoperative complications.Results: Compared with the group Q, the cumulative sufentanil consumption were significantly lower in group QF at 6-12, 12-18, 18-24, and 24h (49.29±16.76 vs. 31.42±18.81μg, P <0.001) after surgery. The postoperative pain intensity was lower in group QF at rest after 6, 12, and 24 h (P <0.05) and at activity after 6, 12, 18, and 24 h (P <0.05). Group QF had higher Qor-15 scores at postoperative 24 h and 48 h (P <0.001) and longer duration before the first opioid require via PCIA (P <0.001) postoperatively than group Q. There was no statistically significant difference between the two groups for complications postoperatively of elderly.Conclusions: Our study provides a multimodal, opioid-sparing analgesic regimen for elderly patients in THA. The combination of T-QLB and FICB provides a significant advantage for early postoperative functional recovery. Further studies are required to confirm the minimum effective dose.Trial registration: Chinese Clinical Trial Registry(ChiCTR2000038686).
Let G be a finite group. A subgroup H of G is called weakly s-permutably embedded in G if there is a subnormal subgroup T of G and an s-permutably embedded subgroup H se of G contained in H such that G = HT and H ∩T ≤ H se. The subgroup H is called weakly s-supplemented in G if G has a subgroup K such that HK = G and H ∩ K ≤ H sG , where H sG is the largest s-permutable subgroup of G contained in H. In this paper, we investigate the influence of weakly s-permutably embedded and weakly s-supplemented subgroups on the structure of finite groups. Some recent results are generalized.
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