Groups with normal restriction property by Hung Phi Tong-Viet | Papers by Hung Phi

Groups with normal restriction property Hung P. Tong-Viet Abstract. Let G be a finite group. A subgroup M of G is said to be an NRsubgroup if, whenever K M, then K G ∩ M = K where K G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an N R-subgroup then G is solvable. This gives a positive answer to a conjecture posed in [2]. Mathematics Subject Classification (2000). Primary 20D10; Secondary 20D05. Keywords. solvable groups, maximal subgroups. 1. Introduction All groups considered are finite. Let G be a group. Following Berkovich in [2], a triple (G, H, K) is said to be special in G if K H ≤ G and H ∩ K G = K, where K G is the normal closure of K in G. A subgroup H is called an N R−subgroup (Normal Restriction) if, whenever K H, then (G, H, K) is special in G. The main result of this paper is a proof of Conjecture 2 raised in [2]. Theorem 1.1. ([2] Conjecture 2) If all maximal subgroups of G are N R-subgroups then G is solvable. In order to prove Theorem 1.1, we need a result on the factorization of almost simple groups. Unfortunately, we cannot avoid using the Classification of Finite Simple Groups in the proof of that result (see Theorem 1.2). Recall that a group G is said to be almost simple if S G ≤ Aut(S) for some non-abelian simple group S. If K is a proper subgroup of G and H is a subgroup of G with K ≤ H < G, then H is called a proper over-group of K in G. Moreover, a subgroup K of G is said to be p-local in G if K = NG (P ) for some non-trivial p-subgroup P of G, p prime. We also say that K is local in G if K is p-local in G for some prime p. This work was completed with the support of University of Birmingham. 2 Hung P. Tong-Viet Finally, a subgroup K of G is said to be local maximal if it is both maximal and local in G. Theorem 1.2. Let S be a non-abelian simple group and S G ≤ Aut(S). Then there exists a non-trivial subgroup K of S such that all proper over-groups of K in S are local in S and G = NG (K)S. The following corollary is used to show that the minimal counter-example to Theorem 1.1 is not simple. Corollary 1.3. Let S be a non-abelian simple group. Then S contains a local maximal subgroup. Proof. Let G = Aut(S) and K be the subgroup of S obtained from Theorem 1.2. Consider the set A of all proper over-groups of K in S. Clearly, A is non-empty and every element of A is a local subgroup of S containing K. The maximum element of A is a maximal subgroup of S and is local. 2. Preliminaries In this section, we collect some results that we need for the proofs of the theorems above. Lemma 2.1. Let K H ≤ G. If H is an N R-subgroup of G then HK G /K G is an N R-subgroup of G/K G . In particular, if K G and all maximal subgroups of G are N R-subgroups, then all maximal subgroups of G/K are also N R-subgroups. Proof. The first statement is Lemma 4(c) in [2]. The second statement follows easily. Theorem 2.2. ([4] Theorem 4.3) Let P be a p-Sylow subgroup of a group G. If P lies in the center of NG (P ) then G has a normal p-complement. Theorem 2.3. ([2] Proposition 7) Let H be a maximal solvable subgroup of G. If H is an N R-subgroup of G then H = G. 3. Proofs of the Theorems Proof of Theorem 1.2. Without loss of generality, we can assume that G = Aut(S). By the Classification of Finite Simple Groups, if S is a non-abelian simple group then S is a finite simple group of Lie type, an alternating group of degree at least 5 or one of 26 sporadic groups. In this proof, we treat the Tits group, 2 F4 (2) as sporadic group rather than a group of Lie type, and in view of the isomorphisms A6 L2 (9), and A5 L2 (5), we consider A5 , A6 to be groups of Lie type. (i) S is a finite simple group of Lie type in characteristic p, S = 2 F4 (2) . By Proposition 8.2.1 and Theorem 13.5.4 in [3], S has a (B,N)-pair. Let B be a Borel subgroup of S. Then B = NS (U ), where U is a p-Sylow subgroup of S. For any Groups with normal restriction property Table 1. |Out(S|) = 1 S K S K M11 J1 M23 M24 2˙S4 7 : 6 23 : 11 24 : A8 Ly Th F i23 Co1 37 : 18 31 : 15 2˙F i22 S3 × A9 Ru 5 : 4 × A5 J4 37 : 12 Co3 2 × M12 B 47 : 23 Co2 210 : M22 : 2 M 2˙B 3 θ ∈ G, as S G, U θ ≤ S θ = S, and hence U θ is a p-Sylow subgroup of S. By Sylow’s Theorem U θ = U g for some g ∈ S. Observe that B θ = NS (U θ ) = NS (U g ) = B g . Thus θg −1 ∈ NG (B), so that θ ∈ NG (B)S, and hence G = NG (B)S. Moreover, if H is any proper over-group of B in S, then H is a parabolic subgroup of S and H < S, so that H is p-local in S. Therefore we can choose K to be a Borel subgroup of S. (ii) S is an alternating group of degree n ≥ 7. In this case G = Sn . Let H = Sn−3 × S3 and K = H ∩ S. Since n − 3 > 3, it follows from [5] that K is a maximal subgroup of S, H is a maximal subgroup of G, and hence G = HS. As [H : K] = 2, we have H = NG (K), so that G = NG (K)S. The subgroup K satisfies the Theorem since it is 3-local and maximal in S. (iii) S is sporadic or S = 2 F4 (2) . By [1], [G : S] = 1 or 2. If G = S then we can choose K to be any local maximal subgroup of S. The pairs (S, K) are given in Table 1. Otherwise, as in (ii), choose H to be a maximal subgroup of G such that K = H ∩S is a local maximal subgroup of S. Then K will satisfy the conclusion of the Theorem. The triple (S, K, H) are given in Table 2. The proof is now completed. Proof of Theorem 1.1. Let G be a minimal counter-example to Theorem 1.1. We first show that G is not simple. By contradiction, suppose that G is simple. By Corollary 1.3, G contains a p-local maximal subgroup M. Let P be a p-subgroup of G such that M = NG (P ). Then 1 = P M and since M is an N R-subgroup of G, we have P G ∩M = P. However as G is simple and P ≤ P G G, P G = G. Hence P = G ∩ M = M. Let P1 be a cyclic subgroup of order p in the center of M. Then G P1 is normal in M. Apply the same argument as above, we have P1 = G, and so G P1 = P1 ∩ M = M. Thus M is a cyclic group of order p. In view of the maximality of M and the simplicity of G, M is a p-Sylow subgroup of G and NG (M ) = M. By Theorem 2.2, G has a normal p-complement. This contradicts to our assumption. Thus G is not simple. Let N be any minimal normal subgroup of G. By Lemma 2.1, the group G/N satisfies the hypothesis of the Theorem and has smaller order than that of G, by the minimality of G, G/N is solvable. Thus N is the unique minimal normal subgroup of G, and it coincides with the last term of the derived series of G. If N is solvable then G is also solvable and we are done. Thus we assume that N is not solvable. 4 Hung P. Tong-Viet Table 2. |Out(S)| = 2 S M12 M22 J2 2 F4 (2) HS J3 M cL He Suz ON F i22 HN F i24 K 42 : D12 24 : A6 A4 × A5 52 : 4A4 5 : 4 × A5 21+4 : A5 − 51+2 : 3 : 8 + 52 : 4A4 35 : M11 . 43 L3 (2) 10 2 : M22 31+4 : 4A5 + . 37 O7 (3) H K ·2 24 : S6 K:2 2 5 : 4S4 5 : 4 × S5 21+4 : S5 − K ·2 52 : 4S4 35 : (M11 × 2) . 43 (L3 (2) × 2) 10 2 : M22 : 2 31+4 : 4S5 + . 37 O7 (3) : 2 Then N = S1 × S2 × · · · × St , where Si = S xi , S is a non-abelian simple group, and x1 , x2 , · · · , xt ∈ G. Let K be the subgroup of S obtained from Theorem 1.2, and T = K1 × K2 × · · · × Kt , where Ki = K xi . Then T is a non-trivial proper subgroup of N. Since N is the unique minimal normal subgroup of G, NG (T ) < G. We will show that G = NG (T )N. For any g ∈ G, since N g = N, there exists a permutation π of degree t acting on {1, 2, · · · , t} such that S xi g = S xiπ . Let gi = xi gx−1 . Then iπ gi ∈ NG (S). We have T g = K x1 g × K x2 g × · · · × K xt g = K g1 x1π × K g2 x2π × · · · × K gt xtπ = = K g1π−1 x1 × K g2π−1 x2 × · · · × K gtπ−1 xt = K h1 x1 × K h2 x2 × · · · × K ht xt = s s s = K x1 s1 × K x2 s2 × · · · × K xt st = K1 1 × K2 2 × · · · × Kt t , where K giπ−1 = K hi with hi ∈ S by Theorem 1.2, and si = hxi ∈ Si . Let i s s s = s1 .s2 . . . st ∈ N. Since [Si , Sj ] = 1 if i = j ∈ {1, 2, . . . , t}, Ki = Ki i . Thus g s T = T , where s ∈ N. Therefore G = NG (T )N. Let M be any maximal subgroup of G containing NG (T ). Let U = M ∩N. We have G = M N, and U = M ∩N M. As G/N = M N/N M/U, M/U is solvable. If U is solvable then M is solvable. By Theorem 2.3, G = M, a contradiction. Thus U is non-solvable. Let L be any non-trivial normal subgroup of M. Since M is maximal in G, M is an N R-subgroup of G, so that L = LG ∩ M. It follows from the fact that N is the unique minimal normal subgroup of G, N ≤ LG . We have U = N ∩ M ≤ LG ∩ M = L. We conclude that U is a minimal normal subgroup of M. Now, since U is a minimal normal subgroup of M and U is non-solvable, U = W1 × W2 × · · · × Wk , where Wi W for all 1 ≤ i ≤ k and W is a non-abelian simple group. Suppose that there exists j ∈ {1, 2, . . . , t} such that Sj ≤ U. As Sj Groups with normal restriction property is normal in N, G N M Sj = Sj M = Sj ≤ M. 5 G However as Sj = N, G = M N = M, a contradiction. Therefore Sj ∩ U < Sj for any j ∈ {1, 2, . . . , t}. Since Kj ≤ Sj N, Kj ≤ Sj ∩U U. As U is a direct product of non-abelian simple groups and Sj ∩ U is a non-trivial normal subgroup of U, there exists a non-empty set J ⊆ {1, 2, . . . , t} such that Sj ∩ U = i∈J Wi . Hence Kj ≤ i∈J Wi < Sj , x−1 j and so K≤ i∈J Wi < S, where Wi x−1 j are non-abelian simple for any i ∈ J. However, by Theorem 1.2, x−1 j i∈J Wi is local in S. This final contradiction completes the proof. References [1] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985, Maximal subgroups and ordinary characters for simple groups, with computational assistance from J.G. Thackray. [2] Yakov Berkovich, Subgroups with the Character Restriction Property and Related Topics, Houston Journal of Mathematics, Vol. 24, No. 4, 1998. [3] Roger W. Carter, Simple Groups of Lie Type, Pure and Applied Mathematics, Vol. 28. John Wiley and Sons, London-New York-Sydney, 1972. [4] Daniel Gorenstein, Finite Groups, Chelsea Publishing Company, Second Edition, 1980. [5] M.W. Liebeck, C.E. Praeger, Jan Saxl, A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups, Jorunal of Algebra 111, 365-383 (1987). Acknowledgment I would like to thank Professor Kay Magaard and Doctor Le Thien Tung for their help with the preparation of this work. I am also grateful to Professor Derek Holt and the referee for their suggestions to improve the proof of Theorem 1.2. Hung P. Tong-Viet School of Mathematics University of Birmingham Edgbaston, Birmingham, B15 2TT United Kingdom e-mail: tongviet@maths.bham.ac.uk