




Volume 36 • Number 1 • 2013 

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Recognition of Finite Simple Groups Whose First Prime Graph Components are $r$Regular
Liangcai Zhang, Wujie Shi, Dapeng Yu and Jin Wang
Abstract.
Let $G$ be a finite group and $\pi(G)=\{p_{1},p_{2},\cdots,p_{s}\}$. For $p\in \pi(G)$, we put $\deg(p):$ $=\{q\in \pi(G)  p\sim q$ in the prime graph of $G$$\}$, which is called the degree of $p$. We also define $D(G):=(\deg(p_{1}),\deg(p_{2}),\ldots,\deg(p_{s}))$, where $p_{1}$ < $p_{2}$ < $\cdots$ < $p_{s}$, which is called the degree pattern of $G$. We say $G$ is $k$fold $OD$characterizable if there exist exactly $k$ nonisomorphic finite groups having the same order and degree pattern as $G$. In particular, a $1$fold $OD$characterizable group is simply called an $OD$characterizable group. In the present paper, we determine all finite simple groups whose first prime graph components are $1$regular and prove that all finite simple groups whose first prime graph components are $r$regular except $U_{4}(2)$ are $OD$characterizable, where $0\leq r\leq2$. In particular, $U_{4}(2)$ is exactly $2$fold $OD$characterizable, which improves an earlier obtained result.
2010 Mathematics Subject Classification: 20D05, 20D06, 20D60
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