View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 210 (2007) 789–796 www.elsevier.com/locate/jpaa On chief factors of finite groups Xiaolei Liua,b,∗, Nanqing Dingb a Department of Mathematics and Physics, Zhongyuan University of Technology, Zhengzhou City, Henan Province, 450007, China b Department of Mathematics, Nanjing University, Nanjing, 210093, China Received 8 October 2005; received in revised form 29 August 2006 Available online 18 December 2006 Communicated by J. Walker Abstract Let H and K be normal subgroups of a finite group G and let K ≤ H. If A is a subgroup of G such that AH = AK or A ∩ H = A ∩ K , we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results. c 2006 Elsevier B.V. All rights reserved. MSC: 20D10; 20D20 1. Introduction Covering and avoidance have proved to be very interesting and useful concepts when characterizing finite solvable groups and some of their subgroups. For example, Gaschutz¨ [3] introduced a conjugacy class of subgroups of a finite solvable group, so-called pre-Frattini subgroups. Pre-Frattini subgroups avoid the complemented chief factors of a finite solvable group G but cover the rest of its chief factors. Chambers [1] got a sufficient condition for a subgroup of a finite solvable group to be an f -pre-Frattini subgroup. Tomkinson [7] gave a general method for constructing subgroups which either cover or avoid each chief factor of a finite solvable group. In these papers, the authors aimed to find subgroups having the covering and avoidance properties of a finite soluble group. In 1993, Ezquerro [2] considered problems converse to above approach and gave some characterization for a finite group G to be p-supersolvable and supersolvable under the assumption that all maximal subgroups of some Sylow subgroup of G have the covering and avoidance properties. In his paper, Guo [4] pushed this approach further and obtained some characterizations of a finite solvable group based on the assumption that some of its maximal subgroups or 2-maximal subgroups have the covering and avoidance properties. We observe that the previous authors imposed strong conditions on a finite group, which confines authors to a very restricted area, to study a finite group. In this paper, we will extend the previous methods to chief factors of finite groups (not necessarily solvable). To be precise, we will investigate the solvability of chief factors of a finite group ∗ Corresponding author at: Department of Mathematics and Physics, Zhongyuan University of Technology, Zhengzhou City, Henan Province, 450007, China. E-mail addresses: [email protected] (X. Liu), [email protected] (N. Ding). 0022-4049/$ - see front matter c 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2006.11.008 790 X. Liu, N. Ding / Journal of Pure and Applied Algebra 210 (2007) 789–796 with the help of covering and avoidance of some subgroups. Fortunately, we get some meaning theorems. And thus we may obtain known results as corollaries of our theorems in many cases. In this paper G always denotes a finite group, p a prime, π(G) the set of prime divisors of the order of G. For the sake of convenience, we call H/K a factor group of a finite group G if H and K are normal subgroups with K ≤ H. 2. Elementary results Definition 1. Let A be a subgroup of a group G and H and K normal subgroups of G. Further suppose H ≥ K . We will say that: (1) A covers H/K if HA = KA. (2) A avoids H/K if H ∩ A = K ∩ A. We will say that A has the covering and avoidance properties in G if A either covers or avoids every chief factor of G. Also we say that A is a CAP-subgroup of G. For convenience, further, we introduce, following [4], the following notation. Let F be the set of maximal subgroups of G. Fn = {M ∈ F and M is non-nilpotent}. Fc = {M ∈ F | |G : M| is composite}. p F = {M ∈ F and NG (P) ≤ M for a Sylow p-subgroup P of G}. op S p F = p∈π(G)−{2} F . ocn op F = F ∩ Fc ∩ Fn. These are families of subgroups of G. Definition 2. Socn = ∩{M ∈ F ocn} if F ocn is non-empty; otherwise Socn(G) = G. We note that Socn(G) is a characteristic subgroup of G, and that Φ(G) ≤ Socn(G) always holds. Lemma 1. (1) Let A be a subgroup of G and H/K a factor group of G. (a) A covers H/K if and only if K (H ∩ A) = H. (b) A avoids H/K if and only if K A ∩ H = K. (2) Let A ≤ G and H/K a chief factor of G. If A ∩ H E G, A covers or avoids H/K . In particular, normal subgroups cover or avoid any chief factor of G. (3) G is a non-abelian simple group if and only if any non-trivial proper subgroup of G is not a CAP-subgroup. (4) Let A be a subgroup of G and H/K a chief factor of G. Then A covers or avoids H/K if and only if there exists a normal subgroup N with N ≤ A ∩ K and A/N covers or avoids (H/N)/(K/N) respectively. Further, A is a CAP-subgroup of G if and only if there exists a normal subgroup L of G which is contained in A and such that A/L is a CAP-subgroup of G/L. (5) Let A be a subgroup of G and H/K a factor group of G. If (|A|, |H/K |) = 1, A avoids H/K. (6) Let H/K be a factor group of G and A a subgroup of G that covers or avoids H/K . Suppose that B ≥ A is a subgroup of G. Then (H ∩ B)/(K ∩ B) is covered or avoided by A as is H/K. Proof. (1) Easy. (2) Since A ∩ H E G,(A ∩ H)K/K E G/K . Noting that (A ∩ H)K/K ≤ H/K and that H/K is a minimal normal subgroup of G/K,(A ∩ H)K/K = H/K or 1, that is, (A ∩ H)K = H or (A ∩ H)K = K , which implies that A covers or avoids H/K . (3) Easy. (4) Let N = (A ∩ K )G . Then AH = AK ⇐⇒ (A/N)(H/N) = (A/N)(K/N) and A ∩ H = A ∩ K ⇐⇒ (A/N) ∩ (H/N) = (A/N) ∩ (K/N). (5) We have the identity |H/K | = |AH : AK ||A ∩ H : A ∩ K |; noting |A ∩ H : A ∩ K | | |A|, |A ∩ H : A ∩ K | = 1, this shows A ∩ H = A ∩ K , that is, A avoids H/K . (6) Easy. Lemma 2. Let H/K be a factor group of G and A a subgroup which covers H/K of G. Further suppose that P is a group theoretical property which is inherited by subgroups and quotient groups. Then H/K has P if A has P. X. Liu, N. Ding / Journal of Pure and Applied Algebra 210 (2007) 789–796 791 Proof. Suppose that A has the property P and that A covers H/K , that is, AH = AK . We have H/K ≤ AH/K = ∼ AK/K = A/A ∩ K . Noting the choice of P, the lemma is proved. Lemma 3. Let H/K be a chief factor of G. Every cyclic subgroup of G covers or avoids H/K ⇐⇒ H/K is of prime order. Proof. H⇒ Suppose p ∈ π(H/K ) and let A/K be a subgroup with order p of H/K . We may assume A/K = hxiK/K , where x ∈ H. Of course, x is not contained in K . By assumption, Hhxi = K hxi, or H ∩ hxi = K ∩ hxi. If Hhxi = K hxi, H/K is of order p by Lemma 2. Suppose H ∩ hxi = K ∩ hxi. But H ∩ hxi = hxi since x ∈ H. On the other hand, K ∩ hxi < hxi because x is not contained in K . This contradiction shows that H/K is of prime order. ⇐H We need only to note the identity |H/K | = |AH : AK ||A ∩ H : A ∩ K | and |H/K | = p for some prime. Proposition 1. Let G be a finite group and let V be a subgroup of G. If V covers the factor group H/K with order a prime power of G, every Hall subgroup of V covers or avoids H/K. Proof. Let Q be a Hall subgroup of V . Set π = π(Q). We prove that Q covers or avoids H/K . By the assumption, we may suppose |H/K | = pα for some prime p ∈ π(G). If p is not contained in π, Q avoids H/K by Lemma 1(5). Suppose p ∈ π. Observing HV = KV, |HV |π = |KV |π . Also |HV |π = |HQ|π , |KV |π = |KQ|π . Thus |HQ|π = |KQ|π . So |HQ : KQ| = |HQ|/|KQ| = (|HQ|π |HQ|π0 )/(|KQ|π |KQ|π0 ) = |HQ|π0 /|KQ|π0 , which is a π 0-number. On the other hand, |HQ : KQ| | |H/K | = pα. Hence |HQ : KQ| = 1, that is, HQ = KQ; in other words, Q covers H/K . Remark. Without the solvability of H/K , the above theorem is false.