Geodesic
Finite simple groups whose Sylow $2$-subgroups contain a~cyclic subgroup of index~$16$
Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 181-203

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In the paper the following main result is proved. Theorem.Let $X$ be a finite simple group with Sylow $2$-subgroup $P$. Suppose that $P$ has a cyclic subgroup of index $16$. Then either the sectional $2$-rank of $X$ does not exceed $4$, or $|P|\leqslant2^8$, or $X\cong L_2(32)$. A use of results of Gorenstein and Harada (RZh.Mat., 1975, 5A192), Kondrat'ev (RZh.Mat., 1977, 12A192), Beisiegel (RZh.Mat., 1977, 12A191) and Volker Stingl leads to the conclusion that finite simple groups whose $2$-subgroups have a cyclic subgroup of index $16$ are known. Bibliography: 24 titles. 
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     author = {\`E. M. Pal'chik},
     title = {Finite simple groups whose {Sylow} $2$-subgroups contain a~cyclic subgroup of index~$16$},
     journal = {Sbornik. Mathematics},
     pages = {181--203},
     publisher = {mathdoc},
     volume = {37},
     number = {2},
     year = {1980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1980_37_2_a2/}
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È. M. Pal'chik. Finite simple groups whose Sylow $2$-subgroups contain a~cyclic subgroup of index~$16$. Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 181-203. http://geodesic.mathdoc.fr/item/SM_1980_37_2_a2/