UDC 512.54 Let G ~ be the image of a finite group G under a homeomorphism ~ with the kernel ker ~ = ~ lying in the Frattini subgroup O(G) of the group G. In this case we shall call G a Frattini extension of the group G ~. Obviously a Frattini extension is an unsplittable extension of the group r by the group G ~. Different constructions of unsplittable extensions of elementary abelian groups by the classical groups were studied in [1-4]. In the present article, as was done by the author in [1] for the groups PSL (2, 2P), we construct Frattini extensions of the groups PSL (n, pY) except for the case p = 2. We remark that the group �9 here, as in [1], is not necessarily abelian. On the other hand Frattini extensions are of interest because they preserve many properties of the original group. Among such properties, for example, are nilpotence and solvability (cf. [5, Theorems III.3.6 and III.3.7]), and under certain additional hypotheses the property of being a minimal non-9"-group, where 5 is a local formation (cf. [6, Theorem 24.4]). In particular G is a minimal nonsolvable group, i.e., a nonsolvable group all of whose proper subgroups are solvable, if and only if G ~ is a minimal nonsolvable group. If a5 = ~(G) in this case, then G ~ is a minimal simple group, and by a result of Thompson [7], it is isomorphic to Sz (2P), PSL (2,p), PSL (2,2P), PSL(2, 3P), where p is some prime number, or PSL (3,3). Thus for suitable n, p, and f, the group studied in the present article is a minimal nonsolvable group. 1. Some definitions and statement of the main result. We define a Frattini p-extension to be a Frattini extension of G for which the kernel �9 is a p-group. In this case we set 01 = 4~, ~i+1 = [Oi, ~]4~ for i > 1. The subgroup 4)i+1 is the intersection of all G-normal subgroups H such that the section Oi/H is an elementary abelian p-group lying in the center of the group ~/H. Therefore we can define an action of the group G ~ on the section 1//= Oi/Oi+l as conjugation by the corresponding elements of the quotient group G/~2i+l. Thus 1~ becomes a GF (p) [G~]-module. We shall call it the ith Frattini G~-module in the group G. The sequence of such modules Vi, i = 1, 2,..., will be called the Frattini G~-series of the group G, and the largest natural number k for which Vk ~ 0 will be called the Frattini G~-length of the group G. The radical rad V of the module V is defined as the smallest submodule of V for which the quotient module is semi-simple. Galois rings play an important role in the present paper. Let R~ = (Z/(pd))[x] be the ring of polynomials over the ring of residues modulo pd. The Galois ring R = GR(pyd,p 4) is the quotient ring of R1 over the ideal generated by a monic polynomial of degree f that is irreducible modulo the maximal ideal J of the ring R. The properties of Galois rings will be discussed below. In particular the ring R is local and R/J ~- GF (pY). Theorem. Let n >_ 2, let f and d >_ 2 be natural numbers, and let p be a prime; also, if p = 2, then let n = 2 and f > 2. The group G = SL(n, GR(p/d,pd)) is a Frattini p-extension of the group G ~' = SL(n, GF (pY)). Here the following assertiona hold: