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Let G be a finite simple group whose order is of the form pm where p is a prime, (p,m)-1, and the index of a Sylow /^-subgroup in its normalizer is three in G. Suppose the degree equation for the principal />-b!ock, B 0 (p), has the form 1+ 2 a = 3*5 C + 2*3 e 5 / where a, b, c, d, e and / are non-negative integers. In this paper it is shown that under these conditions G must be isomorphic to one of the groups L(2,7), C/(3,3), L(3,4) and A s. This is accomplished by solving the exponential Diophantine degree equation for B Q (p).

No abstract

In this paper all Solutions to the equation l + χ -f y = z, where ;c, y and z are positive integers such that xyz has the form 2 r 3 s 7 r , with r, 5-and t non-negative integers, are determined. This work extends earlier work of the authors and J. L. Brenner in the field of exponential Diophantine equations.

ABSTRACT. In this paper we complete the solution to the equation tu + x + y = z, where w,z,y asid 2 are positive integers and wxyz has the form 2r3s5t with r, s and 1 non-negative integers. Here we consider the case