Let G be a finite group, and let p be an odd prime of |G| and Sylow p -subgroups of G be non-abelian. In this paper, we prove that any two p -subgroups of equal order in G are conjugate if and only if G/O_{p'}(G) is isomorphic to {}^{2}F_{4}(2) and p=3 , \mathrm{Ru} and p=3 , J_{4} and p=3 , \mathrm{Th} and p=5 , or an almost simple group of the socle {}^{2}F_{4}(2^{2n+1}) with n > 0, n\not\equiv 1\bmod 3 and p=3 . This solved a problem of Brandl’s.

In this paper we investigate the optimal problem for the L_{p} mixed chord integral. The existence of the L_{p} chord integral–Petty bodies is established, and the L_{p} geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.

We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like weak and strong Lefschetz. As a consequence, we get a different construction of André’s category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases.

We propose a definition of an equivariant algebraic enhanced de Rham functor, which is defined by using the enhanced de Rham functor due to D’Agnolo and Kashiwara. Moreover, as a small application of this functor, we give an approach to the proof of the well-known fact that any equivariant algebraic coherent \mathcal{D} -module is regular holonomic in the case that the number of orbits is finite.

We deal with pseudo-spherical surfaces admitting certain singularities and their caustics. In particular, we give characterizations of cuspidal butterfly, cuspidal lip and cuspidal beak singularities on a pseudo-spherical surface. Moreover, characterizations of certain singularities on the caustics of a pseudo-spherical surface are given. Furthermore, when the caustic has a cuspidal edge singularity, we investigate geometric invariants defined at that point.

Let R be an arbitrary ring and \mathcal{E} an injectively resolving class of left R -modules. We prove that the class of \mathcal{E} -Gorenstein flat right R -modules is closed under extensions, and hence projectively resolving. This answers an open question in Gao and Zhong [Rocky Mountain J. Math. 54 (2024), 143–160] affirmatively. As a consequence, we get that this class is covering. In addition, we introduce the notion of \mathcal{E} -projectively coresolved Gorenstein flat modules, and prove that the class of \mathcal{E} -projectively coresolved Gorenstein flat right R -modules is projectively resolving and closed under transfinite extensions.

The concept of “multi-microlocalization” was introduced to extend the usual microlocal sheaf theory to a more general scope. This paper aims to further extend this theory by exploring advanced topics. One is a stalk formula for multi-microlocalized Hom functors and we compute some examples in multi-microlocal settings. Secondly we construct the Sato triangle in the context of multi-microlocal analysis. Ultimately we get the Sato triangle for the multi-microlocalized Hom functors.

Let R be a commutative noetherian local ring and denote by \operatorname{mod}R the category of finitely generated R -modules. In this paper, we give some evaluations of the singular locus of R and annihilators of Tor and Ext from a viewpoint of the finiteness of dimensions/radii of full subcategories of \operatorname{mod}R . As an application, we recover a theorem of Dey and Takahashi when R is Cohen–Macaulay. Moreover, we obtain the divergence of the dimensions of specific full subcategories of \operatorname{mod}R in the non-Cohen–Macaulay case.

A subgroup H of a group G is permutably complemented if it admits a permutable complement, that is, a subgroup K of G such that G= HK and H \cap K =\{1\} . A group G in which every subgroup is permutably complemented is said to be a C -group. The main purpose of this paper is to investigate the behaviour of uncountable groups of regular cardinality \aleph in which all subgroups of cardinality \aleph are permutably complemented, with the particular aim of identifying sufficient conditions for such groups to be C -groups. Finally, in such a context, uncountable soluble groups of cardinality \aleph are considered by restricting attention to only subnormal subgroups of cardinality \aleph whose subnormal defect is at most 2 .