Abstract: | It is proved that the permutation wreath product H of a simple Suzuki group Sz(27) and a subgroup fo a symmetric group of degree 23, isomorphic to a Frobenius group of order 253, is (up to isomorphism) distinguished among all finite groups by the set of orders of its elements. Since H possesses a minimal normal subgroup N that contains an element of order equal to the exponent of N, this result furnishes a counterexample to one of the conjectures set forth by Shi [1]. In addition, we show that the direct square of a group Sz(27) is also distinguished by the set of orders of its elements. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 304–322, May–June, 1997. |