Geodesic
Thompson's conjecture for some finite simple groups with connected prime graph
Algebra i logika, Tome 51 (2012) no. 6, pp. 683-721

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Let $n$ be an even number and either $q=8$ or $q>9$. We prove a conjecture of Thompson (Problem 12.38 in the Kourovka Notebook) for an infinite class of finite simple groups of Lie type. More precisely, if $S\in\{C_n(q),B_n(q)\}$, then every finite group $G$ for which $Z(G)=1$ and $N(G)=N(S)$ will be isomorphic to $S$. Note that $N(G)=\{n\colon G$ has a conjugacy class of size $n\}$. The main consequence of this result is showing the validity of $AAM$'s conjecture (Problem 16.1 in the Kourovka Notebook) for the groups under study.
Mots-clés : simple group, conjugacy class
Keywords: minimal normal subgroup, centralizer. 
@article{AL_2012_51_6_a0,
     author = {N. Ahanjideh},
     title = {Thompson's conjecture for some finite simple groups with connected prime graph},
     journal = {Algebra i logika},
     pages = {683--721},
     publisher = {mathdoc},
     volume = {51},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2012_51_6_a0/}
}
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N. Ahanjideh. Thompson's conjecture for some finite simple groups with connected prime graph. Algebra i logika, Tome 51 (2012) no. 6, pp. 683-721. http://geodesic.mathdoc.fr/item/AL_2012_51_6_a0/