arXiv subjects
Publications and source records attributed to Jiping Zhang.
In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type $\mathsf A$, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.
By the reduction theorems of Navarro--Tiep and Späth, a way to prove the Alperin weight conjecture and its blockwise version is to verify the co-called inductive Alperin weight condition and inductive blockwise Alperin weight condition for all finie simple groups respectively. In this paper, we establish the inductive blockwise Alperin weight condition for simple groups of type $\mathsf B$ and odd primes, using a criterion given by Brough and Späth recently.
In this paper, we get an elementary and important lemma(See Lemma 3.2) which is about pushout and pullback of modules. And we prove a weak form of a long open conjecture on vanishing of group cohomology for blocks.
The aim of this paper is to apply character properties of Frobenius group to a local block form of an group algebra. We start by establishing a block form of Brauer permutation Lemma by using block participation of conjugate classes of a group $G$. Then we can define a pair of Frobenius corresponding blocks between a group $G$ and its normal subgroup $N$. A near group condition is given to determine a pair of Frobenius corresponding blocks. With a pair of Frobenius corresponding blocks, we study its group structure. At last we prove connections between nilpotent properties and Frobenius corresponding blocks.
In this paper, we focus on a question of M. Newman on isomorphic subgroups of solvable groups. We get a reduction theorem of this question: for each prime q, assume that this question holds for every characteristic q-groups, then this question holds for every finite solvable groups. Using this reduction theorem, we get some partial answers about this question.
We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for a subgroup $H$, often the normalizer of a $p$-subgroup of $G$. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.
Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.
In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.
Let $G$ be a finite group and $N\unlhd G$. In this note, we construct a class poset of $G$ for some cyclic subgroup $C$ of $G$. And we find a relation between $m_{G,N}$ and the Euler characteristic of some nerve spaces of these posets(see Main Theorem).
Let $G$ be a finite group and $N\unlhd G$ with $|G: N|=p$ for some prime $p$. In this note, to compute $m_{G,N}$ directly, we construct a class poset $\mathfrak{T}_{C}(G)$ of $G$ for some cyclic subgroup $C$. And we find a relation between $m_{G,N}$ and the Euler characteristic of the nerve space $|N(\mathfrak{T}_{C}(G))|$ (see the Theorem 1.3). As an application, we compute $m_{S_5, A_5}=0$ directly, and get $S_5$ is a $B$-group.
Bouc proposed the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$-group $β(G)$ is nilpotent. And he has prove that this conjecture holds when $G$ is solvable. In this paper, we consider the case when $G$ is not solvable. Let $S$ be a nonabelian simple group except the Chevalley groups $A_{n}(q)$, $D_{n}(q)$, $E_{6}(q)$, and $^2A_{n}(q)$, if there exists only one factor of $G$ which is isomorphic to $S$, then $β(G)$ is not solvable, of course, is not nilpotent. That means we prove the conjecture in these cases.
In this paper, we find some exotic fusion systems which have non-trivial strongly closed subgroups, and we prove these fusion systems are also not realizable by p-blocks of finite groups.
We prove that the alternating groups of degree at least $5$ are uniquely determined up to an abelian direct factor by the degrees of their irreducible complex representations. This confirms Huppert's Conjecture for alternating groups.
Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.