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R,,, 6 V', then R~, ee (VAF°). Indeed, ifR~; 6 VAF0 ° and e(R~;) = R~;NF = R~,, then A0 C R,g, A0 C R~;, A~ C_ R,g, and m (R~,)[A0] C m(R,~), but rn0 .... ,rnn E m (R~,) and, consequently, 1 = ~ rnial 6 m(R.,L), i<_n an impossibility. The proof is complete. REFERENCES 1. Yu. L. Ershov, "Boolean families of valuation rings," Algebra Logika, 31, No. 3, 274-294 (1992).2. P. Roquette, "Principal ideal theorem for holomorphy rings in fields," J. Reine Ang. Math., 262/263, 361-374 (1973). Translated by O. Bessonova MINIMAL PERMUTATION REPRESENTATIONS OF FINITE SIMPLE CLASSICAL GROUPS. SPECIAL LINEAR, SYMPLECTIC, AND UNITARY GROUPS V. D. MazurovUDC 519.44A faithful permutation representation of least degree for a group G is called a minimal permutation representation of G. A classical group is a general linear, or symplectic, or unitary, or orthogonal group over a field; a simple classical group is one isomorphic to the (unique) nonabehan composition factor of some classical group.Minimal permutation representations of finite sporadic simple groups have been much studied to date: their degrees and subdegrees are calculated, stabilizers of one and two points are found (see the summary table in [1]). The goal of this paper is to add to this study by treating finite simple classical groups.The degrees of minimal permutation representations of finite classical groups were found by Cooperstein in [2]. Some of the degrees listed therein are, however, not minimal, with inaccuracies arising from the violation of some of the inequalities for small fields or dimensions. Fortunately, the errors were readily amenable to corrections, and a revised list of minimal degrees appeared in [3, p. 175]. Unfortunately, neither is this list free of errors, namely, for groups U2,,,(2), where m is not divisible by 3, and for Pf~+,,(3) = O+,n(3), m > 4, the minimal degrees are indicated incorrectly.

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Mazurova [i] showed that Thompson's sporadic simple group [2] hasapropersubgroup //of minimal index 143,127,000, isomorphic to an extension of the simple group~# (2) by an automorphism of order three. Moreover she computed the character of the representation o£ Thompson's group on the cosets of /~ , and in particular proved that the corresponding permutation group has rank ii. In this paper, we determine the subdegrees and the double stabilizers of this permutation group. Table 1 gives the values of some of the complex characters of the group /~=~#~).Jon all the conjugacy classes of /~, If C is a conjugacy class of elements of ~/ which is not contained in ~/~ then C ~ denotes the union of the class ~ and the class of the inverses of elements o~ ~.For each character ~i in the table, there exist two further irreducible characters of the same degree, namely %f% and ~i~ , where % is an irreducible linear character of ~. Table 2 lists the maximal subgroups of the group /~. Note that each of these is the normalizer of some prime subgroup of H, This subgroup is described in the column headed "normalizer." Furthermore, we shall use the following facts about the normalizers of some of the prime subgroups ~=~Z: