1-form symmetries of 4d N=2 class S theories

We determine the 1-form symmetry group for any 4d4d\mathcal{N}=2𝒩=2 class S theory constructed by compactifying a 6d6d\mathcal{N}=(2,0)𝒩=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d4d, modulo screening and flavor charges. Complete specification of a 4d4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.

On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups

Let $G$ be a group and $H$ be a subgroup of $G$. $H$ is said to be $\mathcal{M}$-normal supplemented in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H_1K<G$ for every maximal subgroup $H_1$ of $H$. Furthermore, $H$ is said to be $\mathcal{M}$-normal embedded in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$ or $H\cap K$ is $\mathcal{M}$-normal supplemented in $G$. In this paper, some new criteria for a group to be nilpotent and $p$-supersolvable for some prime $p$ are obtained.

Action of PGL(2,q) on the Cosets of the Centralizer of an Eliptic Element

Most researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroups. In this paper, we consider the action of centralizer of an elliptic element which is a non maximal subgroup . In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is q and the subdegrees are and .

The Non-Commuting, Non-Generating Graph of a Nilpotent Group

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

On a two-fold cover 2.(2⁶˙G₂(2)) of a maximal subgroup of Rudvalis group Ru

The Schur multiplier M(Ḡ1) ≅4 of the maximal subgroup Ḡ1 = 2⁶˙G₂(2)of the Rudvalis sporadic simple group Ru is a cyclic group of order 4. Hence a full representative group R of the type R = 4.(2⁶˙G₂(2)) exists for Ḡ1. Furthermore, Ḡ1 will have four sets IrrProj(Ḡ1;αi) of irreducible projective characters, where the associated factor sets α1, α2, α3 and α4, have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover 2. Ḡ1 of Ḡ1 which can be treated as a non-split extension of the form Ḡ = 27˙G2(2). The ordinary character table of Ḡ will be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of Ḡ and as well as the sizes of the sets |IrrProj(Hi; αi)| associated with each inertia factor Hi. From the ordinary irreducible characters Irr(Ḡ) of Ḡ, the set IrrProj(Ḡ1; α2) of irreducible projective characters of Ḡ1 with factor set α2 such that α22= 1, can be obtained.

On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups

Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

𝒫-characters and the structure of finite solvable groups

Let 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.