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FINITE LINEAR GROUPS CONTAINING AN IRREDUCIBLE SOLVABLE NORMAL SUBGROUP

The following theorem is proved. Let G be a finite group which has a faithful representation X of degree n over the complex number field such that XI H is irreducible where H is a solvable normal subgroup of G. Let p be a prime and assume that n is neither a multiple of p nor a multiple of a prime power q8 with q8- ? 1 mod p. Then a p-Sylow subgroup of G is normal and abelian.