arXiv subjects
Publications and source records attributed to Heguo Liu.
Let $p$ be an odd prime and $S$ a nonabelian finite $p$-group. In [9, 10], they proposed the following conjecture: if $\mathcal{F}$ be a transitive fusion system over a finite $p$-group $S$, then $S$ is either extraspecial of order $p^{3}$ or elementary abelian. In this note, we use an easy method to prove that this conjecture holds when the $p$-rank of $S$ is 2.
Let $p$ be an odd prime and let $\mathcal{F}$ be a fusion system over a finite $p$-group $P$. A fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. In this paper we provide new criteria for saturated fusion systems $\mathcal{F}$ to be nilpotent, which can be viewed as extension of the $p$-nilpotency theorem of Glauberman and Thompson for fusion systems attributed to Kessar and Linckelmann.
Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times \mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n, m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$, $\Omega_l(V(\mathbb{F}_pG))$ and $\Omega_l(G)$ are given. Furthermore, there is a direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$.
Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the unitary subgroup of $V(FG)$. If $p$ is odd, then the order of $V_*(FG)$ is $|F|^{(|G|-1)/2}$. However, the case when $p=2$ still is open. In this paper, the order of $V_*(FG)$ is computed when $G$ is a nonabelian $2$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{2^n}\times \mathbb{Z}_{2^m} \longrightarrow G \longrightarrow \mathbb{Z}_2\times \cdots\times \mathbb{Z}_2 \longrightarrow 1$$ and $G'\cong \mathbb{Z}_2$, $n, m\geq 1$. Further, a conjecture is confirmed, namely, the order of $V_*(FG)$ can be divisible by $|F|^{\frac{1}{2}(|G|+|\Omega_1(G)|)-1}$, where $\Omega_1(G)=\{g\in G\ |\ g^2=1\}$.
The adjoint of a matrix in the Lie algebra associated with a matrix algebra is a fundamental operator, which can be generalized to a more general operator $\varphi_{AB}: X\rightarrow AX-XB$ by two matrices $A$ and $B$. The kernel of the operator is very well-known and it can be found in Gantmacher's book. The formulas for the dimensions of the kernels of arbitrary powers of the operator $\varphi_{AB}$ were given in terms of the Segre characteristics of these two matrices by the second and third authors in this paper and their collaborators. This paper provides an alternative approach to this problem via the Weyr characteristic in a more essential method. We obtain formulas for the dimensions of the kernels of arbitrary powers of the operator in terms of the Weyr characteristics. Furthermore, the basis for kernel of each power of the operator is described explicitly. As a consequence, for arbitrary square matrices $A$ and $B$ over an algebraically closed field, the dimension of the kernel of each power of the operator $\varphi_{A-\lambda I,B}$ for eigenvalues $\lambda$ of $\varphi_{AB}$ can be viewed as a similarity invariant of the operator $\varphi_{AB}$, so we characterise the operator within similarity, which should be of interest to a number of people (including physicists).
In this paper we define the notion of local rank for fusion systems so as to reformulate the Alperin's weight conjecture in the framework of block fusion systems following the work by Kn\"orr and Robinson.
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.
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.
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.
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.