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\iteman{io-port 05956929}
\itemau{Jiang, Qinhui; Shao, Changguo; Guo, Xiuyun; Shi, Wujie}
\itemti{On Thompson's conjecture of $A_{10}$.}
\itemso{Commun. Algebra 39, No. 7, 2349-2353 (2011).}
\itemab
Introduction: Let $G$ be a finite group. A prime graph $\Gamma(G)$ associated with $G$ is defined as follows. The vertices are the primes dividing the order of $G$ and two vertices $p,q$ are adjacent by an edge if and only if $G$ contains an element of order $pq$. We denote the number of the connected components of $\Gamma(G)$ by $t(G)$. In 1987, Thompson posed the following conjecture. Thompson's Conjecture. Let $G$ be a finite group with $Z(G)=1$ and $M$ a non-Abelian simple group satisfying that $cs(G)=cs(M)$, then $G\cong M$, where $cs(G):=\{n\in\Bbb N\mid G$ has a conjugacy class of size $n\}$. After the fourth author of this article opened the above conjecture in 1989 [see {\it W.-J. Shi} and {\it J.-X. Bi}, Lect. Notes Math. 1456, 171-180 (1990; Zbl 0718.20009)], {\it G.-Y. Chen} has proved that Thompson's Conjecture is true for all simple groups with $t(G)\ge 3$ and almost all simple groups with $t(G)=2$ [see J. Algebra 185, No. 1, 184-193 (1996; Zbl 0861.20018)]. To our knowledge, little is known about whether Thompson's Conjecture is true or not for a finite simple group $G$ with $t(G)=1$. In this short article, we prove that Thompson's Conjecture is true for $A_{10}$, where $A_{10}$ is the alternating group of degree $10$ with $t(A_{10})=1$, that is, we prove the following theorem. Main Theorem. Let $G$ be a finite group and $Z(G)=1$. If $cs(G)=cs(A_{10})$, then $G\cong A_{10}$.
\itemrv{~}
\itemcc{}
\itemut{conjugacy classes; finite simple groups; prime graph components; alternating group $A_{10}$; conjugacy class sizes}
\itemli{doi:10.1080/00927872.2010.488677}
\end