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Let H be a subgroup of a finite group G. We say that H is weakly ss-supplemented in G if there exists a subgroup T of G and an s-semipermutable subgroup Hss of G contained in H such that G = HT and H∩T ≤ Hss. In this paper, we get some new characterizations of supersolvability and nilpotentcy of G by assuming some minimal subgroups of G are weakly ss-supplemented. Some recent results are extended and generalized.

Background The aim of this study was to construct a model based on the prognostic features associated with epithelial–mesenchymal transition (EMT) to explore the various mechanisms and therapeutic strategies available for the treatment of metastasis and invasion by hepatocellular carcinoma (HCC) cells. Methods EMT-associated genes were identified, and their molecular subtypes were determined by consistent clustering analysis. The differentially expressed genes (DEGs) among the molecular subtypes were ascertained using the limma package and they were subjected to functional enrichment analysis. The immune cell scores of the molecular subtypes were evaluated using ESTIMATE, MCPcounter, and GSCA packages of R. A multi-gene prognostic model was constructed using lasso regression, and the immunotherapeutic effects of the model were analyzed using the Imvigor210 cohort. In addition, immunohistochemical analysis was performed on a cohort of HCC tissue to validate gene expression. Results Based on the 59 EMT-associated genes identified, the 365—liver hepatocellular carcinoma (LIHC) samples were divided into two subtypes, C1 and C2. The C1 subtype mostly showed poor prognosis, had higher immune scores compared to the C2 subtype, and showed greater correlation with pathways of tumor progression. A four-gene signature construct was fabricated based on the 1130 DEGs among the subtypes. The construct was highly robust and showed stable predictive efficacy when validated using datasets from different platforms (HCCDB18 and GSE14520). Additionally, compared to currently existing models, our model demonstrated better performance. The results of the immunotherapy cohort showed that patients in the low-risk group have a better immune response, leading to a better patient’s prognosis. Immunohistochemical analysis revealed that the expression levels of the FTCD, PON1, and TMEM45A were significantly over-expressed in 41 normal samples compared to HCC samples, while that of the G6PD was significantly over-expressed in cancerous tissues. Conclusions The four-gene signature construct fabricated based on the EMT-associated genes provides valuable information to further study the pathogenesis and clinical management of HCC.

BACKGROUND Circular RNA (CircRNA) and HBx genes separately play essential roles in the occurrence and development of hepatitis B (HBV)-related hepatocellular carcinoma (HCC). However, whether HBx expression in HCC is co-related to differential circRNA patterns remains unknown. METHODS HCC cell lines with HBx overexpression (HepG2 H6679) and empty vector control (HepG2 H5298) were successfully constructed. The high-throughput second-generation transcriptome sequencing technology (RNA-seq) was employed to sequence the two cell lines, and the selected circRNAs were verified by qPCR (quantitative real-time PCR). The differentially expressed circRNAs were analyzed. Bioinformatics analyses, including clustering, differential expression, GO analysis, and KEGG pathway, were performed. Target Scan and Miranda software were employed to predict miRNAs corresponding with circRNAs. RESULTS We identified 1120 circRNAs upregulated and 1447 circRNAs downregulated in HepG2 cell lines with HBx overexpression compared to its control. We selected 36 circRNAs with significant differences (also consistent with log2fold change absolute value ≥ 1.0 or P ≤ 0.05) displayed by cluster analysis and then performed qPCR validation. Among them, 15 circRNAs (hsa_circ_0005603, hsa_circ_0004448, hsa_circ_0006845, hsa_circ_0064654, hsa_circ_0006460, hsa_circ_0045350, hsa_circ_0000824, hsa_circ_0005227, hsa_circ_0067991, hsa_circ_0064656, hsa_circ_0005224, circRNA11716, circRNA759, circRNA14848 and circRNA13751) are consistent with sequencing results. Hsa_circ_0005603 and hsa_circ_0006845 showed significant differences and were chosen for further study. GO analysis shows that many target genes are involved in biological processes, cellular components, and molecular functions. Nearly 193 target genes were enriched on KEGG pathways analysis. Actin cytoskeleton regulation, tight junction, and FoxO signaling pathway are among the top three pathways involved in most genes. We predicted that hsa_circ_0005603 might interact with micro-RNAs, including miR-182-5p, hsa-miR-27a-3p, hsa-miR-98-5p, and hsa-miR-198, that might thereby regulate downstream genes involved in tumor progression. Similarly, hsa_circ_0006845 was predicted to be referred to HBV-related HCC by acting as a sponge for hsa-miR-106a-3p and hsa-miR-198. Furthermore, we discovered two novel circular RNAs (circRNA11716 and circRNA13751) which might be involved in HCC occurrence. CONCLUSION In this study, we comprehensively explored the differentially expressed circRNAs in HepG2 cells with different HBx expression, and our results indicate that hsa_circ_0005603, hsa_circ_0006845, and novel circular RNAs (circRNA11716 and circRNA13751) might play an important role in HBV-related HCC, deserving further research.

Hepatocellular carcinoma (HCC) is a common high-mortality cancer, mainly due to diagnostic difficulties during its early clinical stages. In this study, we aimed to identify genes that are important for HCC diagnosis and treatment, and we investigated the underlying mechanism of prognostic differences. Differentially expressed genes (DEGs) were identified by using the limma package, and receiver operating characteristic curve analysis was performed to identify diagnostic markers for HCC. Bioinformatics and clinical specimens were used to assess epithelial cell transforming 2 (ECT2) in terms of expression, prognostic value, pathways, and immune correlations. In vitro experiments were used to investigate the underlying mechanism and function of ECT2, and the results were confirmed through in vivo experiments. The integrated analysis revealed 53 upregulated DEGs, and one candidate biomarker for diagnosis (ECT2) was detected. High expression of ECT2 was found to be an independent prognostic risk factor for HCC. ECT2 expression showed a strong correlation with tumor-associated macrophages. We found that ECT2 overexpression increased the migration and proliferation of HCC cells. It also promoted the expression of PLK1, which subsequently interacted with PTEN and interfered with its nuclear translocation, ultimately enhancing aerobic glycolysis and promoting M2 macrophage polarization. M2 macrophages suppress the functions of NK cells and T cells, and this was confirmed in the in vivo experiments. Overall, ECT2 may promote the polarization of M2 macrophages by enhancing aerobic glycolysis and suppressing the functions of immune cells. ECT2 could serve as a candidate diagnostic and prognostic biomarker for HCC.

We will give generalized definitions called type II n-cocycles and weak quasi-bialgebra and also show properties of type II n-cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras.