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Simple groups with large Sylow subgroups
A. V. Romanovskii
A. I. Kostrikin posed the problem of the structure of a simple group having a Sylow $p$-subgroup $P$ for which $|P|^3>|G|$, and $C(x)\subset PC(P)$ whenever $x\in P^\sharp$. It has been established by the author that $PSL(2,q)$, and $Sz(q)$ are the only simple groups of this kind. Earlier Brauer and Reynolds have found the solution to the problem of Artin which is the partial case of Kostrikin's problem when $|P|=p$. One of the results used in the proof of the main theorem of the author leads to the following group-theoretical characterization of $PSL(2,q)$: a simple group $G$ is isomorphic to $PSL(2,q)$, $q>3$, if and only if $G$ contains a $CC$-subgroup of odd order $m$ distinct from its own normalizer in $G$, and such that $|G|<(m+1)^3$.
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Mathematics of the USSR-Sbornik, 1982, 43:3, 377–393
MSC: 20D05, 20D20
A. V. Romanovskii, “Simple groups with large Sylow subgroups”, Mat. Sb. (N.S.), 115(157):3(7) (1981), 426–444; Math. USSR-Sb., 43:3 (1982), 377–393
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\paper Simple groups with large Sylow subgroups
\jour Mat. Sb. (N.S.)
\jour Math. USSR-Sb.
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A. V. Romanovskii, “A characterization of simple Zassenhaus groups”, Math. USSR-Sb., 47:2 (1984), 397–409
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