Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 163–173. c Indian Academy of Sciences On non-Frattini chief factors and solvability of finite groups JIANJUN LIU1,∗, XIUYUN GUO2 and QIANLU LI3 1School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China 2Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China 3Department of Mathematics, Shanxi Datong University, Datong, Shanxi 037009, People’s Republic of China ∗ Corresponding author. E-mail: [email protected] MS received 16 January 2011; revised 27 April 2011 ∗ Abstract. A subgroup H of a group G is said to be a semi CAP -subgroup of G if there is a chief series 1 = G0 < G1 < ··· < Gm = G of G such that for every non-Frattini chief factor Gi /Gi−1, H either covers Gi /Gi−1 or avoids Gi /Gi−1.In this paper, some sufficient conditions for a normal subgroup of a finite group to be solvable are given based on the assumption that some maximal subgroups are semi ∗ CAP -subgroups. Keywords. Frattini chief factors; solvable groups; semi cover-avoiding properties. 1. Introduction All groups considered in this paper are finite. Let G be a solvable group. It is well-known that every maximal subgroup M of G is a CAP-subgroup of G, that is MH = MK or M ∩ H = M ∩ K for every chief factor H/K of G. The converse of this result is also true. In fact, Guo and Shum [4] proved the following theorem. Theorem A (Theorem 3.1 of [4]). A group G is solvable if and only if every maximal subgroup M of G in F ocn(G) is a C AP-subgroup of G, where F ocn(G) ={M G| and ∈ ( ) = ( ) ≤ , | : | there exists P Sylp G with p 2 such that NG P M G M is composite and M is non-nilpotent }. If we read the proof of Theorem A, it is easy to see that the key point of the proof is the following: Lemma B(Lemma 2.8 of [4]). A group G is solvable if there exists a minimal normal subgroup N of G and a solvable maximal subgroup M of G such that M avoids N/1, that is M ∩ N = 1. It is clear that N/1 in Lemma B is a non-Frattini chief factor of G. So an interesting question is: How about the influence of non-Frattini chief factors on the solvability of finite groups? 163 164 Jianjun Liu, Xiuyun Guo and Qianlu Li In this paper, we use non-Frattini chief factors in a group to investigate the solvability of the group. In §2 we introduce a new concept: semi CAP∗-subgroups, which is related to partly non-Frattini chief factors in a group. Then, in §3, we prove some sufficient condi- tions for a normal subgroup of a group to be solvable provided some maximal subgroups in the group are semi CAP∗-subgroups. 2. Basic definitions and preliminary results In this section, we give the definition of semi CAP∗-subgroups. Then we discuss some properties of semi CAP∗-subgroup. Let K and L be normal subgroups of a group G with L ≤ K . Then K/L is called a normal factor of G. A subgroup H of G is said to cover K/L if HK = HL. On the other hand, if H ∩ K = H ∩ L, then H is said to avoid K/L.IfK/L is a chief factor of G and K/L ≤ (G/L) (respectively K/L (G/L)), then K/L is said to be a Frattini (respectively non-Frattini) chief factor of G. DEFINITION 2.1 A subgroup H of a group G is said to be a semi CAP∗-subgroup of G if there is a chief series 1 = G0 < G1 < ··· < Gm = G of G such that for every non-Frattini chief factor Gi /Gi−1, H either covers Gi /Gi−1 or avoids Gi /Gi−1. Recall that a subgroup H of a group G is said to be a semi CAP-subgroup of G if H covers or avoids every chief factor of some chief series of G (Definition 2.1 of [3]). It is clear that a semi CAP-subgroup of a group G mustbeasemiCAP∗-subgroup of G. However, the converse is not true. Example 2.2. Let P =a ×b be the direct product of two cyclic groups a and b of order 4 and c ∈ Aut(P) such that ac = a2b3, bc = a3b. Then the semidirect product K = P c is of order 24 × 3. We set G = K ×d , the direct product of K and a cyclic subgroup d of order 2. It is easy to see that the Frattini subgroup (G) =a2, b2 of G is a minimal normal subgroup of G, the series 1 <(G)<P < K < G is a chief series of G, (G)/1 is Frattini and the rest are non-Frattini. We can see that H =a2 avoids every non-Frattini chief factors of this series and therefore H is a semi CAP∗-subgroup of G. However, H ∩ (G) = H = 1 = H ∩ 1 and H(G) = (G) = H. This implies that the chief factor (G)/1 is neither covered nor avoided by H. Similarly, (G)C2/C2 is neither covered nor avoided by H, where C2 =d .Onthe other hand, every chief series of G must contain one of (G)/1 and (G)C2/C2. Thus H is not a semi CAP-subgroup of G. Lemma 2.3 (Lemma 1.2.20 of [1]). Let K/L be a chief factor of a group G. If N is a normal subgroup of G contained in L, then K/L is a Frattini chief factor of G if and only if (K/N)/(L/N) is a Frattini chief factor of G/N. Solvability of finite groups 165 Lemma 2.4. Let K/L be a chief factor of a group G and N a normal subgroup of G. Then: (1) If N avoids K/L and K/L ≤ (G/L), then K N/L N is a chief factor of G and KN/LN ≤ (G/LN). (2) If N covers K/L and K/L (G/L), then (K ∩ N)/(L ∩ N) is a chief factor of G and (K ∩ N)/(L ∩ N) (G/L ∩ N). (3) If N covers K/L and N ≤ (G), then K/L ≤ (G/L) and (K ∩ N)/(L ∩ N) ≤ (G/L ∩ N). (4) Let N avoids K/L and N ≤ (G). Then K/L (G/L) if and only if K N/LN (G/LN). Proof. If N avoids K/L, then it follows from KN/LN =∼ K/L that KN/LN is a chief factor of G.IfN covers K/L, then it follows from (K ∩ N)/(L ∩ N) =∼ K/L that (K ∩ N)/(L ∩ N) is a chief factor of G. (1) Since K/L ≤ (G/L), (KN/L)/(LN/L) ≤ ((G/L)/(LN/L)) and therefore KN/LN ≤ (G/LN). (2) By hypothesis, there exists a maximal subgroup M/L of G/L such that K/L M/L. If K ∩ N ≤ M, then L(K ∩ N) = K ∩ LN = K ∩ KN = K ≤ M, a contradiction. Hence (K ∩ N)/(L ∩ N) (G/L ∩ N) (3) Since N ≤ (G), we can see that (K ∩N)/(L∩N) ≤ (G)/(L∩N) ≤ (G/L∩N). Also since N ≤ (G), we see that NL/L ≤ (G)L/L ≤ (G/L) and therefore KN/L = NL/L ≤ (G/L). Hence K/L ≤ (G/L). (4) By (1), we only need to prove the necessary condition. If K/L (G/L), then there exists a maximal subgroup M/L of G/L such that KM = G. Since N ≤ (G), M/LN is a maximal subgroup of G/LN and MKN = G. This completes our proof. Lemma 2.5. Let N be a normal subgroup of a group G and N ≤ H. Then H is a semi CAP∗-subgroup of G if and only if H/NisasemiCAP∗-subgroup of G/N. Proof. Suppose that H is a semi CAP∗-subgroup of G. Then, there exists a chief series 1 = G0 < G1 < ···< Gm = G of G such that H covers or avoids every non-Frattini chief factor of this series. It is easy to see that the following series: 1 = G0 N/N ≤ G1 N/N ≤ G2 N/N ≤···≤ Gm N/N = G/N (∗) is a chief series of G/N. Suppose that (Gi N/N)/(Gi−1 N/N) is a non-Frattini chief factor of G/N. For finishing our proof, we only need to prove either HGi N = HGi−1 N or H ∩ Gi N = H ∩ Gi−1 N. Since N G, NGi = NGi−1 or N ∩ Gi = N ∩ Gi−1. Clearly HGi N = HGi−1 N if NGi = NGi−1. So in the following we may assume that N ∩ Gi = N ∩ Gi−1. By Lemma 2.3, we have Gi N/Gi−1 N (G/Gi−1 N). Applying Lemma 2.4(1), we see Gi /Gi−1 (G/Gi−1) and therefore H covers or avoids Gi /Gi−1.IfH covers Gi /Gi−1, then it follows from HGi = HGi−1 that HGi N = HGi−1 N. Hence we may assume that H avoids Gi /Gi−1, that is H ∩ Gi = H ∩ Gi−1. This implies that H ∩ Gi N = (H ∩ Gi )N = H ∩ Gi−1 N, as desired. 166 Jianjun Liu, Xiuyun Guo and Qianlu Li Conversely, if H/N is a semi CAP∗-subgroup of G/N, then there exists a chief series ¯ 1 = G0/N < G1/N < G2/N < ···< Gm/N = G/N of G/N such that H/N covers or avoids every non-Frattini chief factor of this series.