Finite Groups with Weakly $$\mathcal {H}C$$ H C -Embedded Subgroups
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Finite Groups with Weakly HC-Embedded Subgroups Qinghong Guo1 · Xuanli He1
· Jianquan Liang1
Received: 10 September 2019 / Revised: 27 November 2019 / Accepted: 3 December 2019 © Iranian Mathematical Society 2019
Abstract A subgroup H of a finite group G is said to be weakly HC-embedded in G if there exists a normal subgroup T of G such that H G = H T and H g ∩ N T (H ) ≤ H for all g ∈ G, where H G is the normal closure of H in G. In this paper, we investigate the structure of the finite group G under the assumption that P a Sylow p-subgroup of G, where p is a prime dividing the order of G, and we fix a subgroup of P of order d with 1 < d < |P| such that every subgroup H of P of order p n d (n = 0, 1) is weakly HC-embedded in G. Keywords Weakly HC-embedded subgroup · p-Supersoluble · p-Nilpotent · Saturated formation Mathematics Subject Classification 20D10 · 20D20
1 Introduction All groups considered in this paper will be finite. The terminology and notation employed agree with standard usage as in Doerk and Hawkes [7]. G always denotes a finite group. A class of groups F is a formation provided that (1) if G ∈ F and N G, then G/N ∈ F ; and (2) if G/Ni ∈ F , i = 1, 2, then G/(N1 ∩ N2 ) ∈ F for any normal subgroups Ni of G. A formation F is said to be saturated if G/(G) ∈ F implies that G ∈ F . Let U be the class of all supersolvable groups. Clearly, U is
Communicated by Mohammad Reza Darafsheh.
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Xuanli He [email protected] Qinghong Guo [email protected] Jianquan Liang [email protected]
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College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
a saturated formation. The U -hypercenter of a group G will be denoted by Z U (G), that is, the product of all the normal subgroups N of G whose G-chief factors below N have prime order. For more details about saturated formations, see [7], I V ]. A normal subgroup N of G is said to be hypercyclically (resp., p-hypercyclically) embedded in G if every G-chief factor (resp., p-chief factor) of G below N is cyclic (see [15], 5.2). If G/N is supersoluble (resp., p-supersoluble), then G is supersoluble (resp., psupersoluble) if and only if N is hypercyclically (resp., p-hypercyclically) embedded in G. For any group G, the generalized Fitting subgroup F ∗ (G) is the set of all elements x of G which induces an inner automorphism on every chief factor of G. Clearly, F ∗ (G) is a characteristic subgroup of G (see [12], X , Sect. 13). We shall use F p (G) which denotes the p-Fitting subgroup of G. In fact, F p (G) = O p p (G). In [17], Wang, in 1996, introduced the concept of c-normality as follows: a subgroup H of G is said to be c-normal in G if there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ HG , where HG is the normal core of H in G, that is, the largest normal subgroup of G contained in H . Many authors use this concept to study the structure of finite groups(see [2,5,17]). Another subgroup embedding property was introduced by Bianch
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