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Order components of a finite group are introduced in Chen (J.
Algebra 185 (1996) 184). It was proved that $PSL(3, q)$, where $q$
is an odd prime power, is uniquely determined by its order
components (J. Pure and Applied Algebra (2002)). Also in
Iranmanesh and et al. (Acta Math. Sinica, English Series (2002))
and (Bull. Austral. Math. Soc. (2002)) it was proved that $PSL(3,
q)$ for $q = 2^{n}$ and $PSL(5, q)$ are uniquely determined by
their order components. Also it was proved that $PSL(p, q)$ is
uniquely determined by its order components (Comm. Algebra
(2004)). In this paper we discuss about the characterizability of
$PSL(p + 1, q)$ by its order component(s), where $p$ is an odd
prime number. In fact we prove that $PSL(p + 1, q)$ is uniquely
determined by its order component(s) if and only if $(q -1)|(p+
1)$. A main consequence of our results is the validity of
Thompson's conjecture for the groups $PSL(p + 1, q)$ where $(q -
1)|(p + 1)$.
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