Abstract:
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 $|G|<(m+1)^3$.
Bibliography: 28 titles.
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