Simple groups contain minimal simple groups

Publicacions Matem`atiques, Vol 41 (1997), 411–415. SIMPLE GROUPS CONTAIN MINIMAL SIMPLE GROUPS Michael J. J. Barry and Michael B. Ward Abstract It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group. Aminimal simple group is a non-abelian simple group all of whose proper subgroups are solvable [7]. The minimal simple groups were classified by Thompson in [7, Corollary 1]. It follows from the definition that every finite non-solvable group Gcontains a subgroups Hand K with KHand H/K a minimal simple group. When Gis simple we show that Kcan be chosen to be {e}. More precisely our result is Theorem 1. If Gis a finite non-abelian simple group then Gcontains a subgroup which is a minimal simple group. In some ways, the result seems obvious. What is surprising is that it does not appear in print anywhere and that some experts do not believe it at first glance. Before we give the proof of the theorem we present a lemma which will help us deal with the groups of Lie type. This lemma was pointed out to us by Gary Seitz. Lemma 1. Let qbe a power of a prime. Suppose Ghas a subgroup Hwhich is isomorphic to either SL3(q)or PSL3(q). Then either G is a minimal simple group or Gcontains a proper subgroup which is non-abelian simple. Proof of Lemma 1: If q= 2 or 3, then SL3(q)∼ =PSL3(q) is minimal simple. In this case if H=G, then Gis minimal simple; otherwise G properly contains the simple subgroup H. 412 M. J. J. Barry, M. B. Ward We may assume q≥4. If qis a power of 2, SL3(q) contains, in an obvious way, a copy Kof the simple group SL2(q). Since K∩Z(SL3(q)) = {I}, it follows that PSL3(q) contains a copy of SL2(q) as well. Assume qis odd. Now Ω3(q)≤SL3(q) since the determinant of any orthogonal transformation is ±1 and so any element of the commutator subgroup has determinant 1. Because Ω3(q) is a simple group isomorphic to PSL2(q) it follows that Ω3(q)∩Z(SL3(q)) = {I}. Hence PSL3(q) also contains acopyofΩ 3(q)∼ =PSL2(q). This concludes the proof of Lemma 1. Proof: We will show that if Gis not minimal simple then Gcontains a proper subgroup which is non-abelian simple. The result then follows by induction on the order of G. By the classification of finite simple groups, Gis an alternating group, a group of Lie type, or a sporadic group. Now the alternating group on five symbols is minimal simple and is contained in every alternating group of degree greater than five. The sporadic groups and the Tits’ group are dealt with in the following table. We use [1] to find a maximal subgroup containing a simple group. Group Maximal Subgroup Group Maximal Subgroup M11 L2(11) M12 M11 M22 A7M23 M22 M24 M23 J2A5 Suz A7HS M22 McL M22 Co3HS Co2McL Co1Co2 He S4×L3(2) Fi22 S10 Fi23 S12 Fi 24 Fi23 HN A12 Th U3(8) : 6 BTh M S3×Th J1L2(11) ON J1 J3L2(19) Ly G2(5) Ru A8J4U3(11) : 2 2F4(2)L2(25) So we can assume Gis simple of Lie type. If Gis a Chevalley group then Ghas an associated root system. If the root system contains a Minimal simple groups 413 subsystem of type A2then Gcontains a subgroup which is isomorphic to either SL3(q)orPSL3(q) and we are done by Lemma 1. (Note in a root system of type G2the long roots form a subsystem of type A2.) This only leaves the Chevalley groups PSL2(q), q≥4 and PSp4(q), q≥3. By a theorem of Dickson [2, Hauptsatz 8.27], if PSL2(q) is not minimal simple then it contains a non-abelian simple subgroup isomorphic to either A5or PSL2(r) where rdivides q. If we choose the short root in a base of type C2then the subgroup of PSp(4,q) generated by the root subgroups corresponding to this short root and its negative is isomorphic to PSL2(q) which is simple if q≥4. The group PSp4(3) is isomorphic to PSU4(22) which we will show below contains the simple group PSL2(4). If Gis a Steinberg group then Gis one of 2An(q2), n≥2, 2Dn(q2), n≥4, 2E6(q2)or3D4(q3). Now if we consider two adjacent roots on the Dynkin diagram of Dnor E6which are left fixed by the graph automorphism of order two we see that 2Dn(q2) and 2E6(q2) contain a proper subgroup isomorphic to either SL3(q)orPSL3(q). One way to deal with 3D4(q3) is to consult the list of its maximal subgroups in [4]. Another way is via the following argument pointed out to us by Gary Seitz. If α2is the root in the Dynkin diagram of type D4fixed by the graph automorphism of order three then {±α2,±α1+α2+α3+ α4,±α1+2α2+α3+α4}is a root sytem of type A2which is fixed by the graph automorphism. Hence 3D4(q3) contains a proper subgroup which is isomorphic to either SL3(q)orPSL3(q). If n≥5 then 2An(q2) contains a proper subgroup isomorphic to either SL3(q2)orPSL3(q2). We see this by considering a pair of adjacent roots at one end of the Dynkin diagram of type Anand the adjacent pair at the other end that they are mapped to by the graph automorphism of order 2. The Steinberg groups remaining for consideration are 2An(q2) for n=2,3 and 4. First of all assume that qis a power of 2. For n= 3 or 4, it is clear that 2An(q2) contains a subgroup isomorphic to the simple group SL2(q2). We see this by looking at the roots at the ends of the Dynkin diagram of type An.Now 2A2(22) is not simple. So assume qis even with q≥4 and n= 2. Now the group generated by the centers of a pair of opposite 2-Sylow subgroups is isomorphic to the simple group SL2(q). Assume now that qis odd. We will use the identification of 2An(q2) with PSUn+1(q2). Since PSU3(32) contains the simple group PSL2(7) we will assume q>3 when n= 3. We will exhibit an embedding of Ω+ n(q)inSUn(q2) for n=3,4,5. (Much more is true —see [5, p. 142].) One way to see this is to take an orthonormal basis {w1,w 2,... ,w n}for a non-degenerate n-dimensional hermitian space W over Fq2with hermitian form BW. Let Vbe the span of {w1,w 2,... ,w n}over Fq. The restriction BVof BWto Vmakes Vinto a non-degenerate quadratic 414 M. J. J. Barry, M. B. Ward space over Fq.NowVhas maximal index. This is true when nis 3 or 5 because all odd-dimensional non-degenerate quadratic spaces over Fqhave maximal index; when n= 4 we see that the form BVhas discriminant 1(F∗ q)2equal to the discriminant of the orthogonal sum of two hyperbolic planes. Now map σ∈Ω+ n(V) to the unique τ∈SUn(W) with τ(wi)=σ(wi) for 1 ≤i≤n. When nis odd, Ω+ n(V) has trivial center, whereas Z(Ω+ 4(q)) = {±I}.Because Z(Ω+ n(q)) ⊆Z(SUn(q2)), it follows that PSUn(q2) contains properly the simple group PΩ+ n(q). Finally we consider the groups of Suzuki and Ree. If Gis 2F4(22m+1) then Gcontains a proper subgroup isomorphic to the simple group SL2(22m+1). We see this by looking at the roots at the ends of the Dynkin diagram of type F4which are paired. If Gis 2B2(22m+1), then Gis minimal simple if 2m+ 1 is prime; otherwise Gproperly contains the simple group 2B2(2p) for any prime divisor pof 2m+1 [6]. The centralizer of an involution in 2G2(32m+1) has the form Z2×PSL2(32m+1) [3]. This concludes the proof of the theorem. Acknowledgement. This work was done while the first author was on sabbatical at the University of Oregon. He thanks the Department of Mathematics at Oregon for its wonderful hospitality, Allegheny College for its generous support during the sabbatical, and Gary Seitz for a number of helpful conversations. References 1. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson,“Atlas of finite groups,” Clarendon Press, Oxford, 1985. 2. B. Huppert,“Endliche Gruppen I,” Springer-Verlag, Berlin, 1967. 3. Z. Janko and J. G. Thompson, On a Class of Finite Simple Groups of Ree, J. Algebra 4(1966), 274–292. 4. P. B. Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q) and their automorphism groups, J. Algebra 115 (1988), 182–199. 5. P. Kleidman and M. Liebeck,“The Subgroup Structure of the Finite Classical Groups,” Cambridge University Press, Cambridge, 1990. 6. M. Suzuki, On a class of doubly transitive groups, Ann. of Math 75 (1962), 105–145. Minimal simple groups 415 7. J. G. Thompson, Nonsolvable groups all of whose local subgroups are solvable: I-VI, Bull. Amer. Math. Soc. 74 (1968), 383–437; Pacific J. Math. 33 (1970), 451–536;, 39 (1971), 483–534;, 48 (1973), 511–592;, 50 (1974), 215–297;, 51 (1974), 573–630. Michael J. J. Barry: Department of Mathematics Allegheny College Meadville, PA 16335 U.S.A. e-mail: m[email protected] Michael B. Ward: Department of Mathematics Bucknell University Lewisburg, PA 17837 U.S.A. e-mail: [email protected] Rebut el 21 de Mar¸c de 1996