L7（3）与GL7（3）的OD-刻画

对于任意一个有限群G,令π（G）表示由它的阶的所有素因子构成的集合.构建一种与之相关的简单图,称之为素图,记作Γ（G）.该图的顶点集合是π（G）,图中两顶点p,g相连（记作p～q）的充要条件是群G恰有pq阶元.设π（G）={P₁,p2,…,p_x}.对于任意给定的p∈π（G）,令deg（p）:=|{q∈π（G）|在素图Γ（G）中,p～q}|,并称之为顶点p的度数.同时,定义D（G）:=（deg（p₁）,deg（p₂）,…,deg（p_s））,其中p₁2<…

OD-Characterization of L7（3） and GL7（3）

If $G$ is a finite group, we denote by $\pi(G)$ the set of all the prime factors of the order of $G$' and construct a simple graph $\Gamma(G)$, called the prime graph of $G$, as follows. The vertices of $\Gamma(G)$ are $\pi(G)$ and two distinct vertices $p$, $q$ are joined by an edge, denoted by $p\sim q$, if and only if there is an element of order $pq$ in $G$. For any given $p\in \pi(G)=\{p_{1}, p_{2},\cdots , p_{s}\}$, define ${\rm deg}(p):=|\{q\in \pi(G)\mid q\sim p \ \text{in} \ \Gamma(G)\}|$, which is called the degree of $p$. Define $D(G):=({\rm deg}(p_{1}), {\rm deg}(p_{2}), \cdots,{\rm deg}(p_{s}))$, where $p_{1}