Finite Groups Whose Maximal Subgroups Are Solvable or Have Prime Power Indices
- PDF / 391,040 Bytes
- 5 Pages / 612 x 792 pts (letter) Page_size
- 23 Downloads / 207 Views
nite Groups Whose Maximal Subgroups Are Solvable or Have Prime Power Indices W. Guo1,2,∗ , A. S. Kondrat’ev3,∗∗ , N. V. Maslova3,4,∗∗∗ , and L. Miao5,∗∗∗∗ Received April 23, 2020; revised May 15, 2020; accepted May 25, 2020
Abstract—It is well known that all maximal subgroups of a finite solvable group are solvable and have prime power indices. However, the converse statement does not hold. Finite nonsolvable groups in which all local subgroups are solvable were studied by J. Thompson (1968). R. Guralnick (1983) described all the pairs (G, H) such that G is a finite nonabelian simple group and H is a subgroup of prime power index in G. Several authors studied finite groups in which every subgroup of non-prime-power index (not necessarily maximal) is a group close to nilpotent. Weakening the conditions, E. N. Bazhanova (Demina) and N. V. Maslova (2014) considered the class Jpr of finite groups in which all nonsolvable maximal subgroups have prime power indices and, in particular, described possibilities for nonabelian composition factors of a nonsolvable group from the class Jpr . In the present note, the authors continue the study of the normal structure of a nonsolvable group from Jpr . It is proved that a group from Jpr contains at most one nonabelian chief factor and, for each positive integer n, there exists a group from Jpr such that the number of its nonabelian composition factors is at least n. Moreover, all almost simple groups from Jpr are determined. Keywords: finite group, maximal subgroup, prime power index, nonsolvable subgroup.
DOI: 10.1134/S0081543820040069 1. INTRODUCTION Throughout the paper, we consider only finite groups, and henceforth the term “group” means “finite group.” Our notation and terminology are mostly standard and can be found in [6, 7, 9, 12]. Denote by Soc(G) the socle of a group G (i.e., the subgroup of G generated by its nontrivial minimal normal subgroups). Recall that G is called almost simple if Soc(G) is a nonabelian simple group. It is well known that a group G is almost simple if and only if there exists a nonabelian simple group S such that Inn(S) G ≤ Aut(S); moreover, since here Inn(S) ∼ = S, we will identify S and Inn(S) and write S G ≤ Aut(S). It is well known that all maximal subgroups of a solvable group are solvable and have prime power indices. However, the converse statement does not hold: for example, all maximal subgroups of the nonabelian simple group P SL2 (7) are solvable and have prime power indices. In 1983, 1
Hainan University, Haikou, 580000 China University of Science and Technology of China, Hefei, 230026 China 3 Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 4 Ural Federal University, Yekaterinburg, 620000 Russia 5 Yangzhou University, Yangzhou, 225009 China e-mail: ∗ [email protected], ∗∗ [email protected], ∗∗∗ [email protected], ∗∗∗∗ [email protected] 2
S47
S48
GUO et al.
Guralnick [10, Theorem 1] described all the pairs (G, H) such that G is a nonabelian simpl
Data Loading...
