Let G be an affine algebraic group scheme over a field k . We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical Rad u ( G ) of G . We investigate some properties of Rad u ( G ) , and study those G for which Rad u ( G ) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k -groups.

A subgroup S of a finite group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. A subgroup A of a finite group G is called a p-CAP-subgroup of G if it covers or avoids every p-chief factor of G, and A is called a strong p-CAP-subgroup of G if for any subgroup H of G containing A, A is a p-CAP-subgroup of H. We use the concepts of p-sylowizers and (strong) p-CAP-subgroups to obtain new criteria for p F -hypercentral embedding of normal subgroups.

The Minimal Faithful Permutation Degree of Groups Without Abelian Normal Subgroups

Abstract Suppose that , are elements of a finite group lying in conjugacy classes of coprime sizes. We prove that is an abelian normal subgroup of and, as a consequence, that if and are ‐regular elements for some set of primes , then is a ‐regular conjugacy class in . The latter statement was previously known for ‐separable groups and this generalisation permits us to extend several results concerning the common divisor graph on ‐regular conjugacy classes, for some prime .

We give a description of the Linnell division ring of a countable residually (poly- Z \mathbb {Z} virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group G G with coefficients in this Novikov ring implies the existence of a normal subgroup N ⩽ G N \leqslant G such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) and G / N G/N is poly- Z \mathbb {Z} virtually nilpotent. As a consequence, we show that if G G is an RPVN group of finite type, then its top-degree ℓ 2 \ell ^2 -Betti number vanishes if and only if there is a poly- Z \mathbb {Z} virtually nilpotent quotient G / N G/N such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) . In particular, finitely generated RPVN groups of cohomological dimension 2 2 are virtually free-by-nilpotent if and only if their second ℓ 2 \ell ^2 -Betti number vanishes, and therefore 2 2 -dimensional RPVN groups with vanishing second ℓ 2 \ell ^2 -Betti number are coherent. As another application, we show that if G G is a finitely generated parafree group with c d ( G ) = 2 cd(G) = 2 , then G G satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.

Identifying Malignant Transformation Risk of Dysplastic Oral Lesions Using the S100A7 Biomarker Signature-Based Assay.

Apolipoproteins play important roles in the metabolism of triglyceride-rich lipoproteins. Ketone monoester β-hydroxybutyrate (KEβHB) has been shown to reduce the circulating levels of remnant cholesterol and triglycerides. However, the mechanisms behind this action remain unknown. To investigate the effect of KEβHB supplementation on apolipoproteins and to study whether circulating levels of triglycerides play a role in this effect. The study was a randomized placebo-controlled trial, registered at https://www.clinicaltrials.gov/ (NCT03889210). It included 18 adults (12 men and 6 women) with prediabetes (defined as per the American Diabetes Association criteria). Following an overnight fast, participants ingested a KEβHB or a placebo beverage in a cross-over manner. Serial blood samples were collected from baseline to 150 min at intervals of 30 min. The endpoints were changes in apolipoprotein (apo) A-I, apo B, apo B-48, apo C-II, apo C-III, and apo E. Area under the curve (AUC) analyses were calculated to estimate changes in the studied apolipoproteins over time. Participants were further stratified into 'hypertriglyceridemia' and normal triglyceride levels subgroups. Ingestion of the KEβHB beverage led to a significantly higher AUC for apo C-II (p = 0.023) in the overall cohort. No statistically significant differences in AUCs were found for the other studied apolipoproteins. The subgroup analysis showed significantly lower levels of apo B (and higher levels of apo C-II) after the KEβHB beverage in individuals with hypertriglyceridemia only. No significant associations were found for the other studied apolipoproteins in either subgroup. Exogenously induced acute ketosis resulted in a significantly elevated apo C-II compared with the placebo. Further, the levels of apo B were significantly lowered following ingestion of the KEβHB beverage only among individuals with hypertriglyceridemia. Acute nutritional ketosis may be considered as a potential approach to reduce atherogenic triglyceride-rich lipoproteins in individuals at high cardiovascular disease risk.

Abstract Let be a finite group, a prime number and a Sylow ‐subgroup of . Recently, Malle, Navarro, and Tiep conjectured that the number of ‐Brauer characters of coincides with that of the normalizer if and only if is normal in . We reduce this conjecture to a question about finite simple groups and prove it for the prime . As a by‐product of our work, we prove a reduction theorem for the blockwise version of Alperin's lower bound on ‐Brauer characters and prove it for 2‐blocks of maximal defect. This improves recent results obtained by Malle, Navarro, and Tiep.

Abstract Background Unexplained biliary duct dilation (CBD/PD) is commonly detected incidentally on imaging studies (US, CT, MRI), yet its clinical significance remains unclear. Elevated liver function tests (LFTs) typically suggest biliary pathology, while normal LFTs complicate diagnosis in non-jaundiced patients. Although endoscopic ultrasound (EUS) may provide valuable insights, its yield and cost-effectiveness in this group are not well established. Aims This systematic review aims to evaluate the yield of significant EUS findings in non-jaundiced patients with unexplained biliary duct dilation and normal LFTs. Methods We conducted a comprehensive literature search across multiple databases, including PubMed, Embase, Medline, Google Scholar, Scopus, the Cochrane Library, and Web of Science, from inception to October 2025. Two independent reviewers screened titles, abstracts, and full texts to identify studies reporting EUS findings in non-jaundiced patients with unexplained biliary duct dilation and normal LFTs. Studies with both normal and abnormal LFTs were included, with data specific to the normal LFT subgroup extracted when available. Results Our review yielded 2,540 unique records, and 15 retrospective observational cohort studies were included in our analysis. These studies involved 1,442 non-jaundiced patients with normal LFTs who underwent EUS for unexplained biliary duct dilation. Malignancy was found in only 1.1% (16/1,442) of patients, primarily pancreatic cancer, which was more frequently observed with isolated pancreatic duct dilation. The pooled diagnostic yield for any pathology was 14.5% (209/1,442), with choledocholithiasis as the most common finding 7.4% (108/1,442). Other benign abnormalities included chronic pancreatitis (1.2%), sludge/microlithiasis (1.3%), diverticulum (0.9%), papillary stenosis (0.7%), and ampullary adenoma (0.6%). Factors associated with dilated biliary ducts included age over 65, female gender, Caucasian ethnicity, post-cholecystectomy status, and narcotic use. However, limited demographic data were available for patients with positive EUS findings. Conclusions Our review indicates that EUS in non-jaundiced patients with unexplained biliary duct dilation and normal LFTs has a modest yield of positive findings at 14.5%, predominantly benign. The yield of critical findings, such as malignancy, is low at only 1.1%. Given these results, a cost-effectiveness analysis is crucial to guide clinical decision-making and optimize patient selection for EUS. Factors related to biliary duct dilation, such as patient age, history of cholecystectomy, and narcotic use, are important considerations. Additional research is needed to validate these findings and develop cost-effective strategies for EUS implementation. Funding Agencies None

On normal subgroups of twisted Chevalley groups over commutative rings

To investigate the relationship between intracranial pressure (ICP), anterior pituitary hormones, and structural brain changes in women with idiopathic intracranial hypertension (IIH). Eighteen women with therapy-refractory IIH underwent lumbar puncture, endocrine assessment, and high-resolution brain MRI. Serum levels of pituitary hormones were correlated with ICP and radiological parameters including pituitary volume, flattening, and optic nerve (ON) and optic nerve sheath (ONS) volume. Group comparisons and partial correlations were used to evaluate associations. ICP showed asignificant positive association with thyroid-stimulating hormone (TSH) levels (r = 0.628, p = 0.016), and asignificant negative association with growth hormone (GH) (r = -0.602, p = 0.023). Regarding structural parameters, only the volume of the right ON showed astrong positive association within the subgroup with elevated ICP (r = 0.90, p = 0.005). Correlations between ONS volumes and ICP in the normal pressure subgroup narrowly missed statistical significance. TSH was the only hormone showing asignificant association, with higher TSH levels relating to larger pituitary volume in the normal ICP subgroup (r = 0.88, p = 0.020), but not in the elevated ICP subgroup. Our exploratory findings suggest potential interactions between ICP, endocrine markers, and structural MRI measures. However, due to the limited sample size and variability in endocrine parameters, the results should be interpreted cautiously and considered hypothesis generating rather than clinically directive. Larger studies are needed to determine whether endocrine MRI associations hold true and whether they have diagnostic or clinical relevance.

Let G be a finite group, N be a nontrivial normal subgroup of G and Irr ( G | N ) be the set of all irreducible characters of G whose kernels do not contain N. By acd Pri , even ( G | N ) and acd nm , even ( G | N ) , we mean the average of all even degree primitive characters in Irr ( G | N ) and the average of all even degree non-monomial characters in Irr ( G | N ) , respectively. In this paper, we show that if 0 < acd Pri , even ( G | N ) < 8 / 3 , then G is solvable. Among the other results, we show the solvability of G when 0 < acd nm , even ( G | N ) < 8 / 3 . Obviously, this result can be considered as a generalization of Taketa’s theorem regarding the solvability of M-groups.

Abstract We introduce a simple equivalence relation on strongly minimal sets in a structure of finite Morley rank, which corresponds, in stability theory, to the non-orthogonality of the associated types. We use it in a group 𝐺 of finite Morley rank to define, for each strongly minimal set 𝑋, two connected normal subgroups M G ⁢ ( X ) M_{G}(X) and W G ⁢ ( X ) W_{G}(X) . When 𝐺 is connected, these subgroups provide a central decomposition of 𝐺 that yields a direct product decomposition of G / Z ⁢ ( G ) G/Z(G) into unidimensional factors, as well as a central decomposition of its derived subgroup into unidimensional subgroups.

Abstract Two groups L 1 L_{1} and L 2 L_{2} are compatible if there exists a finite group 𝐺 with isomorphic normal subgroups N 1 N_{1} and N 2 N_{2} such that L 1 ≅ G / N 1 L_{1}\cong G/N_{1} and L 2 ≅ G / N 2 L_{2}\cong G/N_{2} . In this paper, we give new necessary conditions for two groups to be compatible.

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. Denote by [Formula: see text] the set of irreducible [Formula: see text]-)Brauer characters of [Formula: see text]. Let [Formula: see text] be a normal subgroup of [Formula: see text]. [Formula: see text] is said to be a relative [Formula: see text]-group with respect to [Formula: see text] if for every Brauer character [Formula: see text], there exists a subgroup [Formula: see text] of [Formula: see text] containing [Formula: see text] and [Formula: see text] such that [Formula: see text] and [Formula: see text]. We obtain some results about relative [Formula: see text]-groups in this note.

Let G and A be finite groups of coprime orders, with A acting on G via automorphisms. In this context of coprime action, an A-invariant subgroup H of G is called A C p -normal in G if there exists an A-invariant normal subgroup T of G containing H G such that G = HT and ( H ∩ T ) / H G is a p ′ -group, where H G is the core of H in G and p is a fixed prime dividing the order of G. Under the assumptions of some specific n-maximal A-invariant subgroups having A C p -normality, we establish several criteria for p-solvability of finite groups, where n ∈ { 1 , 2 , 3 } . Some earlier results in the literature were extended.

Classical mathematical methods are insufficient for resolving certain issuesin real-life human problems due to the uncertainty of the data. Researchers from around the world have created innovative mathematical models, like soft and fuzzy set theories, to model the uncertainties that arise in different areas. Jun recently developed a hybrid structure that combined fuzzy and soft set concepts. The hybrid structure principle is appliedto groupoids in this paper, and the properties of hybrid ideals and hybrid subgroupoids in groupoids are also described. Furthermore, the notions of hybrid subgroups, hybrid normal subgroups, and hybrid cosets in a group, as well as their key properties, are discussed. In addition, we show that any member of the collection of hybrid cut sets of a hybrid normal subgroup of a group G is a normal subgroup of G in the traditional sense. Finally, we obtain a finite-group hybrid version of Lagrange’s theorem.

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

This research paper, titled "Foundations of Sub groups and the Subgroup Criterion, "offers a detailed foundational study of one of the most critical structural components in abstract algebra: the subgroup. A subgroup is defined as a subset of a group that maintains the group structure under the inherited binary operation. Understanding these internal structures is essential for classifying groups and proving key theorems. The paper begins by reviewing the four group axioms—closure, associativity, identity, and inverse—before focusing on the core problem: establishing if a subset is a subgroup without checking all four axioms directly. This leads to the central topic: the Subgroup Criterion. We meticulously present and prove the efficiency of the one-step test (for finite groups) and the two-step test (for general groups), which dramatically simplifies the verification process. Illustrative examples are provided, analyzing subsets of both commutative groups, such as the additive group of integers (Z, +), and non-commutative groups, such as the symmetric group S3. Ultimately, this paper formalizes the methodology for discovering the internal architecture of any given group, providing a necessary prerequisite for exploring advanced concepts like cosets, normal subgroups, and homomorphic mappings

Let G be a finite group. A subgroup H of G is called weakly H C − embedded in G if there exists a normal subgroup T of G such that H G = HT and H g ∩ N T ( H ) ≤ H for all g ∈ G , where H G is the normal closuer of H in G. A subgroup H of G has a supersolvable supplement in G if there exists a supersolvable subgroup K of G such that G = HK . In this paper, we investigate the structure of G under the assumption that certain subgroups of G of prime power orders either are weakly H C − embedded in G or have supersolvable supplements in G. Our results improved and generalized some recent results in the literature.