Derived p -Length of a p -Soluble Group with Bounded Indices of Fitting p -Subgroups in Their Normal Closures
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DERIVED p-LENGTH OF A p-SOLUBLE GROUP WITH BOUNDED INDICES OF FITTING p-SUBGROUPS IN THEIR NORMAL CLOSURES D. V. Gritsuk1 and A. A. Trofimuk2,3
UDC 512.542
Let G be a p-soluble group. Then G has a subnormal series whose factors are either p0 -groups or Abelian p-groups. The smallest number of Abelian p-factors in all subnormal series of G of this kind is called the derived p-length of G. A subgroup H of the group G is called a Fitting subgroup if H F (G). The existence of a functional dependence of the estimate of derived p-length of a p-soluble group on the value of indices of the Fitting p-subgroups in their normal closures is established.
1. Introduction In the present paper, we consider only finite groups. All notation and definitions agree with the notation and definitions presented in [1, 2]. The notation Y X means that Y is a subgroup of the group X. A series of subgroups 1 = G0 G1 G2 . . . Gn−1 Gn = G
(1)
is called subnormal if, for any i, the subgroup Gi is normal in Gi+1 . The quotient groups Gi+1 /Gi are called factors of this series. Let p be a prime number. If the order of a group G is a power of the number p (is not divisible by p), then the group G is called a p-group (a p0 -group). In 2006, Monakhov proposed the following definition [3]: Let G be a p-soluble group. Then it has a subnormal series whose factors are either p0 -groups or Abelian p-groups. The smallest number of Abelian p-factors in all subnormal series of the group G is called a derived p-length of the p-soluble group G and denoted by lpa (G). If G is a p-group, then the derived p-length of the group G coincides with its derived length. It is clear that lp (G) lpa (G) for any p-soluble group G. Here, lp (G) is the p-length of the p-soluble group G. Estimates for the derived p-length of a p-soluble group under given restrictions imposed on Sylow p-subgroups were obtained in [4–8]. If H and K are normal subgroups in G such that H K and H/K is a minimal normal subgroup of the quotient group G/K, then H/K is called a chief factor of the group G. The chief factor H/K is called Fitting if the subgroup H is contained in a Fitting subgroup F (G) of the group G. Similarly, a subgroup A of the group G is called a Fitting subgroup if A F (G). In [9], Gasch¨utz proved that the chief factor of maximal order in a soluble group is a Fitting chief factor. The upper bounds of invariants of a soluble group (derived length, nilpotent length, p-length, and chief rank) 1
Pushkin Brest State University, Belarus; e-mail: [email protected]. Skorina Gomel State University, Gomel, Belarus; e-mail: [email protected]. 3 Corresponding author. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 366–370, March, 2020. Original article submitted November 20, 2019. 416
0041-5995/20/7203–0416
© 2020
Springer Science+Business Media, LLC
D ERIVED p-L ENGTH OF A p-S OLUBLE G ROUP WITH B OUNDED I NDICES OF F ITTING p-S UBGROUPS
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depending on the Fitting chief factors and Fitting subgroups were obtain
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