Geodesic
Thompson's conjecture for simple groups with connected prime graph
Algebra i logika, Tome 51 (2012) no. 2, pp. 168-192

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We deal with finite simple groups $G$ with the property $\pi(G)\subseteq\{2,3,5,7,11,13,17\}$, where $\pi(G)$ is the set of all prime divisors of the order of the group $G$. The set of all such groups is denoted by $\zeta_{17}$. A conjecture of Thompson in [Unsolved Problems in Group Theory, The Kourovka Notebook, 17th edn., Institute of Mathematics SO RAN, Novosibirsk (2010), Question 12.38] is proved valid for all groups with connected prime graph in $\zeta_{17}$.
Keywords: finite simple group, Thompson’s conjecture. 
@article{AL_2012_51_2_a1,
     author = {I. B. Gorshkov},
     title = {Thompson's conjecture for simple groups with connected prime graph},
     journal = {Algebra i logika},
     pages = {168--192},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2012_51_2_a1/}
}
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I. B. Gorshkov. Thompson's conjecture for simple groups with connected prime graph. Algebra i logika, Tome 51 (2012) no. 2, pp. 168-192. http://geodesic.mathdoc.fr/item/AL_2012_51_2_a1/