On Soluble Radicals of Finite Groups
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ON SOLUBLE RADICALS OF FINITE GROUPS S. Yu. Bashun1 and E. M. Palchik2
UDC 512.542
Assume that G is a finite group, ⇡(G) = {s} [ σ, s > 2, ⌃ is a set of Sylow σ-subgroups in which one subgroup is taken for each pi 2 σ, and R(G) is the largest normal soluble subgroup in G (soluble radical of G ). Moreover, suppose that each Sylow pi -subgroup Gpi 2 ⌃ normalizes the s-subgroup T (i) 6= 1 of the group G. In this case, we establish the conditions under which s divides |R(G)|.
Introduction Let G be a finite group without soluble normal subgroups. Then the components of a layer L(G) of the group G are either Lie-type groups� Chev(ri ), i = 1, k, or groups from the set Spor, or groups from the set {An , n ≥ 5}. / A5 (2), C3 (2), D4 (2), 2 A3 (2) , then there exists a prime divisor t of order |L| of the Let L 2 Chev(ri ). If L 2 group L that does not divide the order of any proper parabolic subgroup of the group L [4] (Lemma 3). In [2], t was called a minisotropic divisor (or, simply, an m-divisor) of the group L and some properties of the groups involving m-divisors were considered (see also [3], Lemma 6). Let G be a finite group, let ⇡(G) = {s} [ σ, s > 2, and let ⌃0 be a set of Sylow pi -subgroups in which one subgroup is taken for each pi 2 σ. Assume that a Sylow pi -subgroup Gpi normalizes an s-subgroup T (i) 6= 1 for T (i) 6= 1, where m is the number of pairwise different each pi 2 σ. In the present paper, we prove that if m i=1 T (i) subgroups T , then Os (G) 6= 1. 1. Notation, Definitions, and Auxiliary Results We consider only finite groups. In the present paper, we use standard notation and terminology of the contemporary theory of finite groups (see, e.g., [4–9]). Thus, in particular, p, r, pi , ri , s, . . . are prime numbers; ⇡ is a set of different prime numbers (⇡ ⇢ P); ⇡ 0 = P\⇡ is the set of prime numbers that do not belong to ⇡ ; |X| is the number of different elements of a finite set X (the order of the set X); (a, b) is the greatest common divisor of numbers a and b; e(q, t) is the least natural number e such that q e ⌘ 1 (mod t) for (q, t) = 1 and t is a prime number, t > 2 (i.e., t is a primitive prime divisor of the number q e − 1); ⇡(n) is the set of all different prime divisors of the integer n; 1 2
Polotsk State University, Polotsk, Belarus; e-mail: [email protected]. Polotsk State University, Polotsk, Belarus.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 326–339, March, 2020. Original article submitted March 28, 2019; revision submitted January 9, 2020. 370
0041-5995/20/7203–0370
© 2020
Springer Science+Business Media, LLC
O N S OLUBLE R ADICALS OF F INITE G ROUPS
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⇡(X) = ⇡(|X|) for the set X ; xg = g −1 xg. Respectively, Gp is a Sylow p-subgroup of the group G; L(G) (layer of the group G) is the central product of all subnormal subgroups L of the group G with L = L0 and L/Z(G) is a simple non-Abelian group (L is a component of the group G); Spor is the set of 26 simple non-Abelian sporadic groups; Chev is the set of Lie-type groups; Cheva is the subset
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