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).
Download TeX format
