subgroups of soluble type

5 Subgroups of soluble type Consider a subgroup H of a soluble group G. Since every minimal normal subgroup of G is abelian, the following implication holds: If G = MH, M is a minimal normal subgroup of G, and H is a proper subgroup of G,thenH ∩ M = 1. (5.1) It is precisely this property which makes the theory of H-projectors and H- covering subgroups, H a Schunck class, so much easier in the soluble universe than in the general finite one. Salomon, in his Doctoral Dissertation [Sal87], has introduced and studied notions of H-projectors and H-covering subgroups (different from the usual ones) which lead to a theory of these subgroups in arbitrary groups resembling the theory of H-projectors and H-covering subgroups in finite soluble groups. His first basic idea is to give a definition of H-projectors along the following lines: Recall that if H is a class of groups, a subgroup H of a group G is called an H-projector of G if H ∈ S(G) ∩ H and if N  G and HN/N ≤ K/N ∈ S(G/N) ∩ H,thenHN/N = K/N.HereS(X) is the set of all subgroups of a group X. Salomon tries to replace the set S(G) of all subgroups of a group G by suit- able subsets d(G)of S(G) which are such that any element H of d(G) enjoys the above property (5.1); he also tries to develop a theory of the “projectors” so obtained by following the classical approach. It is clear that in order to carry out this, one cannot take just any d(G) ⊆ S(G) satisfying (5.1). First of all, since the definition of H-projector involves not only the group G itself but also its quotients, such sets d(G)havetobe chosen not only for G but at least for all quotients of G; in fact, the classical theory of projectors suggests that it would be reasonable to demand that such a choice be made for all finite groups simultaneously. Put differently, to begin with, one chooses a “subgroup functor” d which associates with any group G a set of subgroups d(G)(werefertoelementsofd(G)assubgroups 205 206 5 Subgroups of soluble type of soluble type) subject, of course, to the condition that (5.1) holds for any G and each H in d(G). Moreover, it is also plausible that a subgroup functor d ought to satisfy certain formal properties such as the following: If H ∈ d(G) and N  G,thenHN/N ∈ d(G/N). Not surprisingly, it turns out that the properties relevant here are closely related to properties of ordinary projectors and covering subgroups. The theory developed in this Chapter is largely the work of Salomon [Sal87] and F¨orster [F¨orb], [F¨ora]. 5.1 Subgroup functors and subgroups of soluble type: elementary properties The purpose of this section is to establish the necessary formal properties of the various functors for subgroups of soluble type. The functors t and t  introduced by Salomon in [Sal87] are studied here. Two similar functors r and r  defined by F¨orster [F¨orb] are also studied. A subgroup functor is a function f which assigns to each group G a pos- sibly empty set f(G) of subgroups of G satisfying θ  f(G)  = f  θ(G)  for any isomorphism θ : G −→ G ∗ . Examples 5.1.1. 1. Functor S: it assigns to each group G the set S(G)of all subgroups of G. 2. Functor S n : it associates with each group G the set S n (G) of all sub- normal subgroups of G. 3. Let p be a prime. Let S p be the function assigning to each group G the set S p (G) of all subgroups U of G containing a Sylow p-subgroup of G; S p is a subgroup functor. 4. Let H be a Schunck class, then Proj H () and Cov H () are subgroup func- tors. Given two subgroup functors e and f, we write e ≤ fife(G) ⊆ f(G)for each group G. Definition 5.1.2. Let f be a subgroup functor. We say that f is inherited if f enjoys the following properties: 1. If U ∈ f(T) and T ∈ f(G), U ≤ T ≤ G,thenU ∈ f(G). 2. If U ∈ f(G) and N  G,thenUN/N ∈ f(G/N ). 3. If U/N ∈ f(G/N), N  G,thenU ∈ f(G). If, moreover, f satisfies 4. U ≤ G and T ∈ f(G) implies U ∩ T ∈ f(U), we say that f is w-inherited. Obviously the functors S and S n are w-inherited. S p is inherited but not w-inherited. 5.1 Subgroup functors and subgroups of soluble type 207 Lemma 5.1.3. Let f be an inherited functor. Then 1. If U ∈ f(G) and N  G,thenUN ∈ f(G). 2. If N is a subnormal subgroup of G and N = N 1  N 2  ···  N k = G is a chain from N to G such that 1 ∈ f(N i ) for all i ∈{2, ,k},then N ∈ f(G). In particular, S n (G) ⊆ f(G) for all groups G if 1 ∈ f(X) for all groups X. Proof. 1. follows from Definition 5.1.2 (2 and 3). 2. By 1, N i ∈ f(N i+1 ) for all i ∈{1, ,k − 1}. Applying Defini- tion 5.1.2 (1), it follows that N ∈ f(G).  Lemma 5.1.4. Let f be a w-inherited subgroup functor. If G is a group, U ∈ f(G) and N is a subnormal subgroup of G such that U ≤ N G (N),thenUN ∈ f(G). In particular, if 1 ∈ f(G),then S n (G) ⊆ f(G). Proof. We argue by induction on |G|.LetN 1 be the normal closure of N in G and, for i>1, denote N i = N N i−1 , the normal closure of N in N i−1 .Since N is subnormal in G, there exists n ≥ 1 such that N = N n . Suppose that G = UN 1 ,thenN 2 = N 1 because U normalises N. Repeating the argument with every N k , it follows that N is normal in G. By Lemma 5.1.3 (1), we have UN ∈ f(G). Therefore we may assume that UN 1 is a proper subgroup of G. By induction, UN ∈ f(UN 1 ). Now UN 1 ∈ f(G) yields UN ∈ f(G)by Definition 5.1.2 (1).  Lemma 5.1.5. If f is a w-inherited subgroup functor and X 1 , X 2 ∈ f(G), then X 1 ∩ X 2 ∈ f(G). Proof. Since f is w-inherited, X 1 ∩X 2 ∈ f(X 1 ). Hence X 1 ∩X 2 ∈ f(G) because X 1 ∈ f(G) and f is inherited.  Definition 5.1.6. Let f be a w-inherited subgroup functor and {X i : i ∈I}⊆ f(G). Then the intersection of all subgroups of G belonging to f(G) and con- taining  i∈I X i is the smallest subgroup of G in f(G) containing  i∈I X i . This subgroup is denoted by X i : i ∈I f and called the f-join of {X i : i ∈I}. Theorem 5.1.7. Let f be a w-inherited subgroup functor. For each group G, f(G) is a lattice under the operations “∩”and“·, · f .” 208 5 Subgroups of soluble type t(G)={H ≤ G : T  S ≤ G, S/T strictly semisimple, then (H ∩ S)T  S} r(G)={H ≤ G : T  S ≤ G, S/T strictly semisimple, H ≤ N G (S), then (H ∩ S)T  S} t  (G)={H ≤ G : T  S sn G, S/T strictly semisimple, then (H ∩ S)T  S} r  (G)={H ≤ G : T  S sn G, S/T strictly semisimple, H ≤ N G (S), then (H ∩ S)T  S} t  (G)={H ≤ G : T  S sn G, S/T simple, then (H ∩ S)T  S} r  (G)={H ≤ G : T  S sn G, S/T simple, H ≤ N G (S), then (H ∩ S)T  S} here strictly semisimple groups are those which can be written as direct products of isomorphic simple groups, while semisimple groups are direct products of not necessarily isomorphic simple groups. If H is a subgroup of G such that H ∈ e(G) for some e ∈{t, r, t  , r  , t  , r  }, we shall say that H is a subgroup of soluble type. The functors t and t  have been introduced by Salomon in his Dissertation [Sal87]. There are certain problems Salomon encounters with these two choices of e: the first of these functors does not really give sets t(G) “large enough” whenever G is “highly non-soluble,” while for t  one of the crucial properties, namely, if H ∈ e(G)andN  G,thenH ∈ e(HN), is missing. Later, F¨orster [F¨orb] overcame these problems by introducing the remaining functors. As will be seen in the next section “the r-functors” enjoy the advantage of producing relevant subgroups of primitive groups. Remarks 5.1.9. 1. If G is soluble, then t(G)=r(G)=t  (G)=r  (G)= t  (G)=r  (G)=S(G). 2. Condition “(H ∩ S)T  S” in the definitions of t  and r  implies that H either covers or avoids the simple section S/T ,thatis,(H ∩ S)T ∈{S, T }. 3. In defining t, r, t  ,r  (t  ,r  , respectively) we might have taken the strictly semisimple (simple) groups S/T as direct products of non-abelian simple groups (as non-abelian simple, respectively). Moreover, since every subnormal subgroup of a direct product of non-abelian simple groups is in fact normal by [DH92, A, 4.13] the condition “(H ∩ S)T  S” can be replaced by “(H ∩ S)T sn S.” 4. If H is a subgroup of G such that H ∈ e(G) for some e ∈{t, r, t  , r  , t  , r  },andN is a normal subgroup of G,thenHN/N ∈ e(G/N). 5. S n ≤ t ≤ t  ≤ t  ,r≤ r  ≤ r  ,t≤ r, t  ≤ r  ,t  ≤ r  . 6. r(G)(r  (G), respectively) is the set of all subgroups H of G with the following property: If T  S ≤ G (T  S sn G, respectively), S/T is strictly semisimple, and H ≤ N G (S) ∩ N G (T ), then (H ∩ S)T  S. Definitions 5.1.8. For any group G, we define the following subgroup func- tors: 5.1 Subgroup functors and subgroups of soluble type 209 Proof. Only a proof for Statement 6 is needed. Let S/T denote a strictly semisimple section of G, with simple component E,say,andletH ≤ N G (S). Then T ∗ ,theD 0 (1,E)-residual of S, is a characteristic subgroup of S con- tained in T with strictly semisimple quotient S/T ∗ contained in D 0 (1,E); in particular, H ≤ N G (T ∗ ). Now Statement 6 follows directly from this observa- tion.  Proposition 5.1.10. Let H and N be subgroups of a group G such that N is quasinilpotent and H normalises N. Suppose that the following condition holds: B  A sn N , A/B strictly semisimple, and H ≤ N G (A) implies that (H ∩ A)B  A. (5.2) Then H ∩ N is subnormal in N. Proof. Proceeding by induction on |G| + |N|, we may clearly assume that G = HN, and hence that N is normal in G. In the case N =1orN is a minimal normal subgroup of G, the claim is immediate from condition (5.2); so without loss of generality, there is a normal subgroup M of G such that 1 = M<N. Taking into account that the class of all quasinilpotent groups is an S n -closed homomorph and that condition (5.2) is inherited by M and N/M, we may apply the inductive hypothesis twice to get that H ∩ M is a subnormal subgroup of M and HM ∩ N is a subnormal subgroup of N; in particular, HM ∩ N is quasinilpotent. Therefore, if HM ∩ N is a proper subgroup of N, then another application of the inductive hypothesis yields that H ∩ N = H ∩(HM∩ N ) is subnormal in HM∩ N, and hence that H ∩ N is subnormal in N . Thus, without loss of generality, N = HM ∩ N ≤ HM and G = HN = HM. Therefore we have: H<G= HM whenever M  G and 1 = M<N. (5.3) Assume that the Fitting subgroup of N ,F(N ), is non-trivial. Then N contains an abelian minimal normal subgroup M of G;sinceN =(H ∩ N)M ,wecan deduce that H ∩ N is normal in N because N centralises M. This is due to the fact that M is a direct product of minimal normal subgroups of M which are central in N because N is quasinilpotent. It remains to deal with the case when F(N)=1.ThenN is a direct product of non-abelian simple groups by Proposition 2.2.22 (3). By Condition (5.2), H ∩ M is normal in M for every minimal normal subgroup of G contained in N. Therefore we obtain: N is a direct product of non-abelian simple groups, and H ∩ M =1, whenever M is a minimal normal subgroup of G contained in N. (5.4) Now, N cannot have two non-isomorphic simple direct factors E and F : otherwise for each X in {E,F} we could find minimal normal subgroups M X of G contained in N with X as composition factor, so by Condition (5.3) and Condition (5.4) both M E and M F should be normal complements of H in G, 210 5 Subgroups of soluble type leading to the contradiction that M E ∼ = (M E ×M F ) ∩ H ∼ = M F . Consequently, N is strictly semisimple, and our claim holds by assumption.  Proposition 5.1.11. The functors t  and t  are inherited. The functor t is w-inherited. Proof. We give a proof for t. 1. Let G be a group and let X and U be subgroups of G such that X ∈ t(U) and U ∈ t(G). Consider a strictly semisimple section S/T of G.Weprovethat (S ∩ X)T is normal in G. Clearly we may assume that S/T is a direct product of non-abelian simple groups. Since U ∈ t(G), it follows that (S ∩ U)T is normal in S. Hence (S ∩ U)/(T ∩ U) is either 1 or a strictly semisimple section of U .IfS ∩ U = T ∩ U,thenS ∩ X = T ∩ X and (S ∩ X)T = T.Thuswemay assume that S ∩ U = T ∩ U .SinceX ∈ t(U ), we have that (S ∩ X)(T ∩ U) is normal in S ∩ U.Thisimpliesthat(S ∩ X)T is normal in (S ∩ U )T  S. Since S/T is a direct product of non-abelian simple groups, we have (S ∩ X)T is normal in S by [DH92, A, 4.13]. Consequently X ∈ t(G). 2. Let U ∈ t(G)andN normal in G. Suppose we have T/N  S/N ≤ G/N and (S/N )  (T/N) strictly semisimple. Then N ≤ T  S ≤ G and S/T is strictly semisimple, so U ∈ t(G) gives that (UN ∩ S)T =(U ∩ S)NT = (U ∩ S)T  S. Therefore UN/N ∈ t(G/N). 3. Consider a normal subgroup N of G.LetU/N ∈ t(G/N). Suppose that S/T is a non-abelian and strictly semisimple section of G.IfTN = SN,thenS =(S ∩ N)T =(S ∩ U)T and (S ∩ U)T is normal in S. Hence we may assume that TN = SN.Inparticular,SN/T N ∼ = S/  (S ∩ N)T ) and (SN/N)  (TN/N) is a strictly semisimple section of G/N.SinceU/N ∈ t(G/N), it follows that  (SN/N) ∩ (U/N)  (TN/N) is normal in SN/N and so (SN ∩ U)T is normal in SN.Thisimpliesthat(SN ∩ U )T ∩ S =(S ∩ U)T is a normal subgroup of S. Therefore U ∈ t(G). The same statements can be obtained for the functors t  and t  just by adding the assumption S subnormal in G for t  ,andS subnormal in G and S/T simple for t  in the above proof. Finally, it is clear that t is w-inherited because every strictly semisimple section of every subgroup of a group is actually a strictly semisimple section of the group itself.  Example 5.1.12. Let E be a non-abelian simple group and let K = E  E be the regular wreath product. Then K can be written as semidirect product K = FE,whereF is the base group and E is its canonical complement in K.ConsidernowG = E  E K, the wreath product of E with K with respect to the transitive permutation representation of K on the right cosets of E. Let B bethebasegroupofG.ThenB = E 1 ×···×E n , n = |K : E|,where E i ∼ = E (i =1, , n), E 1 is E-invariant and E 1 E = E 1 × E. It is rather easy to see that F (B, respectively) is the unique minimal normal subgroup of K (G, respectively). Consequently, G has a unique maximal normal subgroup, namely BF. 5.1 Subgroup functors and subgroups of soluble type 211 Let H be the diagonal subgroup of E 1 E. We claim that H ∈ t  (G). Let T  S sn G with S/T non-abelian and strictly semisimple. If S = G, S is contained in the unique maximal normal subgroup BF of G. Then (by construction of H and BF) H ∩BF = 1, whence H ∩S =1and(H ∩S)T = T  S.IfS = G, then T ∈{G, BF } and (S ∩ H)T = HT = HBF = G = S. Assume, by way of contradiction, that H ∈ r  (HB). Then HN/N ∈ r  (HB/N) whenever N  HB.ButB = E 1 × C,whereC = E 2 ×···×E n and HB/C = (E×E 1 )C/C ∼ = E×E 1 via the canonical isomorphism. This maps the diagonal subgroup H of E × E 1 onto the non-normal subgroup HC/C of EB/C,which contradicts HC/C ∈ r  (EB/C) (see Proposition 5.1.10). Therefore H/∈ r  (HB) and, in particular, H/∈ t  (HB). This shows that t  is not w-inherited. We do not know, however, whether r is w-inherited. Remark 5.1.13. Since t is w-inherited, then, by Theorem 5.1.7, t(G) is closed under the operations “∩”and“·, · t .” In general, U, V  t = U, V :letn ≥ 5 and let G = Σ n be the symmetric group of degree n.LetX 1 = (1, 2) and X 2 = (2, 3).ThenX i ∈ t(G), i =1,2,X 1 ,X 2  ∼ = Σ 3 ,andX 1 ,X 2  t = G, as we can see by looking at the sections Alt(n)/1ofSym(n). Proposition 5.1.14. If H is a proper subgroup of a group G = HM,with H ∈ r  (G),andM is a minimal normal subgroup of G,thenH ∩ M =1. Proof. By Proposition 5.1.10, H ∩ M is normal in M.SinceH is a proper subgroup of G and M is a minimal normal subgroup of G,wehavethat H ∩ M =1.  Remark 5.1.15. The statement of Proposition 5.1.14 does not hold for ele- ments of t  (G). Consider the regular wreath product G = E  T of a non-abelian simple group E and a group T of order 2. Denote by D the T -invariant diag- onal subgroup of the base group B, and put H = TD.ThenG = HB, B is a minimal normal subgroup of G, H ∈ t  (G), but H ∩ B =1. Since the functors t  and r  do not have the crucial property described in Proposition 5.1.14, they are not so interesting for us as the functors t, r, t  , and r  . Nevertheless, we shall see at the end of the section (Theorem 5.1.25) that the composition factors of the subgroups in r  (G) are composition factors of the whole group. Definition 5.1.16. A subgroup functor f is called inductive (respectively, weakly inductive) if it satisfies Conditions 1 and 2 (respectively, 1 and 2  ): 1. G ∈ f(G) 2. If H ≤ K ≤ G, N  G, H ∈ f(K), N ≤ K,andK/N ∈ f(G/N),then H ∈ f(G). 212 5 Subgroups of soluble type 2  . H ≤ G, N  G, H ∈ f(HN),andHN/N ∈ f(G/N) implies that H ∈ f(G). It is clear that w-inherited functors f such that f(X) is non-empty for all groups X are inductive and inductive functors are also weakly inductive. Proposition 5.1.17. t  and t  are inductive functors. Proof. We see that t  is inductive. The same proof applies to t  . First of all, the defining Condition 1 for inductivity of t  holds trivially. To verify Condition 2, let H ≤ K ≤ G and N  G such that N ≤ K, H ∈ t  (K) and K/N ∈ t  (G/N). We show that (S ∩ H)T is normal in G whenever S/T is a non-abelian simple section of G such that S is subnormal in G. Suppose that SN = TN.ThenS =(S ∩ N)T =(S ∩ K)T and (S ∩ K)/(T ∩ K)isa simple section of K such that S ∩ K is subnormal in K.SinceH ∈ t  (K), it follows that (S ∩ H)(T ∩ K) is normal in S ∩ K. Hence (S ∩ H)T is normal in (S ∩ K)T = S. Suppose that SN = TN.Then(SN/N)  (TN/N)isasimple section of G/N and SN/N is subnormal in G/N.SinceK/N ∈ t  (G/N), we have that (K ∩SN)T is normal in SN. Hence (K ∩S)T =  S ∩(K ∩SN)  T = S ∩ (K ∩ SN)T is normal in S ∩ SN = S.Further,ifK ∩ T = K ∩ S,then (S ∩ K)/(T ∩ K) is a simple section of K and S ∩ K is subnormal in K.As H ∈ t  (K), this gives that  H ∩ (K ∩ S)  (K ∩ T ) is normal in K ∩ S.Thisis also true if K ∩ T = K ∩ S. Consequently we obtain: (H ∩ S)T =(K ∩ H ∩ S)(K ∩ T )T  (K ∩ S)T  S. This proves the inductivity of t  .  Note that, in the special case when K = HN, the same argument still works for the functors r, r  and r  ,forH ≤ N G (S)andN  G imply that HN ≤ N G (SN); hence we get: Proposition 5.1.18. r, r  ,andr  are weakly inductive. Moreover, if H ∈ r(G) and L is a subgroup of G containing H,thenH ∈ r(L). Lemma 5.1.19. Let f ∈{t  , r  } and let G be a group. 1. If H is a subgroup of G and N is a soluble subgroup of G normalised by H,thenH ∈ f(HN). 2. Let H ≤ K ≤ G = HN,whereN is a direct product of non-abelian simple groups. If H ∈ f(G) and K ∩ N is normal in N,thenH ∈ f(K). Proof. 1. For a proof of H ∈ f(HN), we may use induction on |G| and the properties of f verified earlier in this section (Propositions 5.1.17 and 5.1.18) to see that without loss of generality 1 = G = HN, N is a minimal normal subgroup of G, H ∩ N =1,andH is a core-free maximal subgroup of G. Hence G is a primitive group of type 1 and N is the unique minimal normal subgroup of G.LetT  S be a non-abelian strictly semisimple section of G 5.1 Subgroup functors and subgroups of soluble type 213 such that S is subnormal in G. Assume that S is a proper subgroup of G. Then S is contained in some maximal normal subgroup M of G.SinceM =1, it follows that N is contained in M and M = N(H ∩M ). Since M is a proper subgroup of G, H ∩M ∈ f(M) by induction. Hence (S ∩ H)T =(S ∩H ∩M)T is normal in S. Therefore we may assume that S = G.ThenT = 1 because otherwise G would be non-abelian and simple. Consequently N is contained in T and so S =(S ∩ H)T . This means that H ∈ f(G). 2. By hypothesis K ∩ N is normal in N,soK ∩ N is a normal subgroup of KN = G .SinceN is a direct product of non-abelian simple groups, it follows that N =(K ∩ N) × M for a normal subgroup M of G.ThenG/M = KM/M ∼ = K. It is clear therefore that H ∈ f(K) because HM/M ∈ f(G/M).  Theorem 5.1.20. Let f ∈{t  , r  } and let G = H F ∗ (G) be a group which is a product of a subgroup H ∈ f(G) and F ∗ (G), the generalised Fitting subgroup of G.IfM is a normal subgroup of G and M is quasinilpotent, then H ∈ f(HM). Proof. We argue by induction on |G|.IfM is soluble, the result follows from Lemma 5.1.19 (1). We may suppose that M is not contained in F = G S , the soluble radical of G.IfF =1,thenF ∗ (G)andM are both direct products of non-abelian simple groups by Proposition 2.2.22 (3). Moreover HM ∩ F ∗ (G)isnormalinF ∗ (G) because H ∩ F ∗ (G)isnormalinF ∗ (G) by Proposition 5.1.10 and [DH92, A, 4.13]. Applying Lemma 5.1.19 (2), we conclude that H ∈ f(HM). Therefore we can suppose that F = 1. By induc- tion HF/F ∈ f  (HF/F)(MF/F)  . This yields H(HM ∩ F )/(HM ∩ F ) ∈ f  HM/(HM ∩ F )  .SinceH ∈ f  H(HM ∩ F )  by Lemma 5.1.19 (1), we con- clude that H ∈ f(HM) by weakly inheritness.  Lemma 5.1.21. Let G = E 1 ×···×E n be a direct product of n copies of a non-abelian simple group E.LetH beasubgroupofG such that H covers or avoids every simple section of G.ThenH is a normal subgroup of G. Proof. We argue by induction on |G|. It is clear we can assume Core G (H)=1. Since either E j ∩ H =1orE j ≤ H for all j ∈{1, ,n}, it follows that E j ∩ H = 1 because H does not contain any normal subgroup of G. Denote N j = X i=j E i ,1≤ j ≤ n. Suppose that G = HN j for each j ∈{1, ,n}.Inparticular,H is a subdirect subgroup of G. Hence H = R 1 ×···×R t ,whereR k ∼ = R l ∼ = E, 1 ≤ k,l ≤ t.Letg = 1 be an element of R 1 . Without loss of generality, we may assume that π 1 (g) =1,whereπ 1 : G −→ E 1 is the projection of G over its first component. Then 1 = π 1 (R 1 ) is a normal subgroup of π 1 (H)=E 1 , and so π 1 (R 1 )=E 1 .Let1= h ∈ E 1 such that g h = g.Sinceg ∈ R 1 ∩ R h 1 ,it follows that R h 1 is contained in H. Assume [h, R 1 ]=1.Then1=π 1 ([h, R 1 ]) = [h, E 1 ]andh ∈ Z(E 1 ). This contradiction shows that [h, R 1 ] =1.Lett be an element of R 1 such that t h = t.Then1= t h t −1 ∈ H ∩ E 1 = 1 (note that if t =(e 1 , e 2 , ,e n ), t h =(e h 1 ,e 2 , ,e n )). 214 5 Subgroups of soluble type This contradiction proves that HN j is a proper subgroup of G for some j ∈{1, ,n}.ThenG/N j is a simple section of G which is not covered by H. Hence H is contained in N j . By induction H is normal in N j and so H is subnormal in G. This implies that H is a normal subgroup of G [DH92, A, 4.13] and the result follows.  Proposition 5.1.22. Let H be a subgroup of G.ThenH ∈ t(G) if, and only if, every simple section of G is covered or avoided by H. Proof. The cover-avoidance property dealt with here is obviously a special case of the defining property of H ∈ t(G). Conversely, let H be a subgroup of G enjoying this cover-avoidance prop- erty, and let T  S ≤ G and S/T strictly semisimple and non-abelian. Let X =(H ∩ S)T/T.ThenX is a subgroup which covers or avoids every simple section of S/T. By Lemma 5.1.21, X is normal in S/T. Consequently H ∈ t(G).  Remark 5.1.23. r is not characterised by the corresponding property: consider onal subgroup H of G:asH is not normal in G, H 5.1.10, yet H covers or avoids any (non-abelian) simple section S/T of G with H ≤ N G (S). In soluble groups, each composition factor of a subgroup is a composition factor of the whole group. It is not true in general. However, this property holds for subgroups of soluble type as the next result shows. We need first a lemma. Lemma 5.1.24. Let G be a group and H ∈ r  (G). Assume that M is a subnormal subgroup of G such that M = X n i=1 E i ,wheretheE i (i =1, , n) are pairwise isomorphic non-abelian simple groups; further assume that H normalises M.Letπ i : M −→ E i (i =1, , n) denote the projection of M over E i .PutI =  i ∈{1, ,n} :(H ∩ M)π i =1  .ThenH ∩ M is subdirect in X i∈I E i . Proof. Note that M/Ker(π i ) is a simple section of G.SinceH ∈ r  (G), it follows that H covers or avoids M/Ker(π i ). If M ∩ H = M ∩ Ker(π i ), we have π i (M ∩ H) = 1. Therefore I =  i ∈{1, ,n} : π i (M ∩ H) =1  =  i ∈{1, ,n} : M =(M ∩ H)Ker(π i )  . It follows that H ∩ M is a subdirect subgroup of X i∈I E i .  Theorem 5.1.25. Let G be a group. Every composition factor of a subgroup H ∈ r  (G) is isomorphic to a composition factor of G. Proof. Proceeding by induction on |G|, we consider a minimal normal sub- group of G.SinceHM/M ∈ r  (G/M), we may assume that our claim holds for H/(H ∩ M)( ∼ = HM/M). If M is abelian, the result follows. Hence we may assume that M is non-abelian. By Lemma 5.1.24, H ∩ M is a subdirect subgroup of a suitable normal subgroup of M.ThenH ∩ M ∈ D 0 (1,E), where E is the composition factor of M.  the direct product of two copies of a non-abelian simple group E and a diag- ∈/ r(G)byProposition [...]... subgroups of soluble type We begin with a result on t -subgroups which may be viewed as a non-existence result: as a consequence of this theorem, in a monolithic primitive group with non-abelian socle the minimal normal subgroup cannot be complemented by a t-subgroup unless the corresponding quotient of the group is soluble In fact, the results on this type of subgroups show that the non -soluble t -subgroups. .. applying [Hup67, V, 8.7], the Sylow 2 -subgroups of K are cyclic or generalised quaternion This contradicts [Hup67, V, 22.9] The next result shows that non -soluble t -subgroups are close to subnormal subgroups Corollary 5.2.6 If H ∈ t(G) and all composition factors of H are nonabelian, then H is subnormal in G 220 5 Subgroups of soluble type Proof Consider a counterexample G of least order Clearly, 1 < H . which associates with any group G a set of subgroups d(G)(werefertoelementsofd(G)assubgroups 205 206 5 Subgroups of soluble type of soluble type) subject, of course, to the condition that (5.1). corresponding quotient of the group is soluble. In fact, the results on this type of subgroups show that the non -soluble t -subgroups share some properties with non -soluble subnor- mal subgroups. This. and subgroups of soluble type: elementary properties The purpose of this section is to establish the necessary formal properties of the various functors for subgroups of soluble type. The functors