Almost simple groups of Lie type and symmetric designs with $λ$ prime

Locally compact groups built up from p-adic Lie groups, for p in a given set of primes

We analyze the structure of locally compact groups which can be built up from p-adic Lie groups, for p in a given set of primes. In particular, we calculate the scale function and determine tidy subgroups for such groups, and use them to recover the primes needed to build up the group.

无穷多个有限单李型群不能由两个素数阶元素不变生成

Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order

本文证明了无穷多个辛群Sp_{2k}(2)不能由两个素数阶元素不变生成，扩展了有限单群生成性质的研究。

几乎所有圈积特征标值可被给定素数整除

Almost all wreath product character values are divisible by given primes

该研究证明，对于任意有限群 $G$ 和素数 $p$，当 $N$ 趋于无穷时，圈积 $G \wr S_N$ 的特征标表中几乎所有条目都能被 $p$ 整除，推广了 Peluse 和 Soundararajan 关于对称群的结果。

On the Prime Graph Question for Almost Simple Groups with an Alternating Socle

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the Prime Graph Question for all almost simple groups with socle $A_n$ if $n \leq 17$.

近似仿射不变性与素数域：格林问题90

Almost Affine Invariance Over Prime Fields: Green Problem 90

研究确定在素数域上，当参数K满足K=o(log p)时，集合A在所有仿射变换下近似仿射不变，解决格林问题90。

几乎所有交错群均可由两个素阶元素不变生成

Almost all alternating groups are invariably generated by two elements of prime order

本研究定量证明绝大多数交错群A_n可由两个素阶元素不变生成，仅存在极少量例外，从而解答了Guralnick等人提出的相关问题。

The structure of almost Abelian Lie algebras

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media - cosmology, crystallography etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics pe…

The inductive McKay-Navarro conditions for the prime 2 and some groups of Lie type

For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reduced the McKay--Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay--Navarro (also c…

Almost Abelian Lie groups, subgroups and quotients

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified a…

素数环的李理想上 $b$-广义斜导子的刻画

A characterization of $b$-generalized skew derivations on lie ideal of a prime ring

本文刻画了素数环的非中心李理想上满足特定恒等式的两个 $b$-广义斜导子的完整结构。

Ideals in Left Almost Semigroups

A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered un…

Jordanable almost Abelian Lie algebras

We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable then we call the almost Abelian Lie algebra Jordanable. The existence of canonical form allows to describe important structural properties of these Lie algebras in explicit terms. Lie subalgebras, ideals, automorphisms and derivations, as well as quadratic Casimir elements are given in explicitly in a suitable basis.

On the Prime Graph Question for Integral Group Rings of 4-primary groups I

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to $\operatorname{PSL}(2, p^f)$ for $f \leq 2$, establishing the Prime Graph Question for the first time for all automorphic extensions of series of simple groups. Using this, we determine exactly how far the so-called HeLP-method can take us for (almost simple) groups having an order divisible by at most $4$ different primes.