Algebra and Logic, Vol. 45, No. 1, 2006 QUASIRECOGNIZABILITY BY THE SET OF ELEMENT ORDERS FOR GROUPS 3 D 4 (q ), FOR q EVEN O. A. Alekseeva UDC 512.542 Keywords: finite group, simple group, set of element orders, quasirecognizability, prime graph. It is proved that if G is a finite group with an element order set as in the simple group 3 D 4 (q), where q is even, then the commutant of G/F (G) is isomorphic to 3 D 4 (q) and the factor group G/G ′ is a cyclic {2, 3}-group. Let G be a finite group. Denote by ω(G) the set of all element orders of G. This set defines a prime graph (the Gruenberg–Kegel graph) GK(G) of G, in which vertices are prime divisors of the order of G and two distinct vertices, p and q, are joined by an edge iff G contains an element of order pq. Denote the number of connected components of the graph GK(G) by t(G), and the set of its connected components by {π i (G) | 1  i  t(G)}; moreover, we assume that 2 ∈ π 1 (G) if G is of even order. The set ω(G) is partially ordered under divisibility, and is uniquely defined by a subset μ(G) of its maximal elements. Below is a structural theorem, which has been proven to hold for finite groups with a disconnected prime graph. THEOREM (Gruenberg–Kegel) [1, Thm. A]. If G is a finite group with a disconnected graph GK(G), then one of the following statements is true: (a) G is a Frobenius group; (b) G = ABC, where A and AB are normal subgroups of G, and AB and BC are Frobenius groups with kernels A and B and complements B and C, respectively. (c) G is an extension of a nilpotent π 1 (G)-group by a group A, where Inn(P ) ≤ A ≤ Aut(P ), P is a simple non-Abelian group with t(G)  t(P ), and A/P is a π 1 (G)-group. Finite simple non-Abelian groups with a disconnected prime graph were described in [1, 2]. Results on finite groups with a disconnected Gruenberg–Kegel graph have found wide application in studies dealing in recognizability of finite groups by sets of element orders (see, e.g., [3]). A finite group G is said to be recognizable (by its element order set) if H ∼ = G for any finite group H with ω(H )= ω(G). The first step in solving the recognizability problem for finite simple groups with a disconnected Gruenberg–Kegel graph is proving the condition for quasirecognizability, which is weaker than recogniz- ability. A finite simple non-Abelian group P is said to be quasirecognizable if any finite group G with ω(G)= ω(P ) has a composition factor isomorphic to P . Note that for a quasirecognizable finite simple group, a question by Shi in [4, Question 12.39], asking whether finite simple groups are recognizable given the set of element orders and order of a group, has been answered in the affirmative. Chelyabinsk Humanitarian Institute, Chelyabinsk, Russia; oksana88888@yandex.ru. Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3-19, January-February, 2006. Original article submitted March 25, 2005; revised July 8, 2005. 0002-5232/06/4501-0001 c  2006 Springer Science+Business Media, Inc. 1