COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS 3 MARTINOGARONZI 1 0 2 Abstract. Givenafinitenon-cyclicgroupG,callσ(G) thesmallestnumber n ofpropersubgroupsofGneededtocoverG. LucchiniandDetomiconjectured a thatifanonabeliangroupGissuchthatσ(G)<σ(G/N)foreverynon-trivial J normalsubgroupN ofGthenGismonolithic,meaningthatitadmitsaunique minimalnormalsubgroup. In thispaper weshow how thisconjecture canbe 4 attacked bythedirectstudyofmonolithicgroups. ] R G Every group considered in this paper is assumed to be finite, unless specified . otherwise. h Given a non-cyclic groupG, call σ(G) - the covering number ofG - the smallest t a number of proper subgroups of G whose union equals G. It is an easy exercise to m showthatσ(G)>2(i.e. nogroupistheunionoftwopropersubgroups). Notethat [ there always exist minimal covers consisting of maximal subgroups. The covering numberhasbeenintroducedthefirsttimebyCohnin1994[Cohn]. Weusuallycall 1 v cover of G a family of proper subgroups of G which covers G, and minimal cover 3 of G a cover of G consisting of exactly σ(G) elements. If G is cyclic then σ(G) is 4 not well defined because no proper subgroup contains any generator of G; in this 7 case we define σ(G)= , with the convention that n< for every integer n. 0 ∞ ∞ 1. Remark 1. If N is a normal subgroup of a group G then σ(G) σ(G/N): indeed, ≤ 0 every cover of G/N can be lifted to a cover of G. 3 1 Given a family of subsets of a group G which covers G, we say that is H H : “irredundant”if K =G for everyH . Clearly everyminimal coveris v H∋K6=H 6 ∈H i irredundant, butSthe converse is false. Actually the notion of irredundant cover is X much weaker than that of minimal cover: for example, if n 2 is an integer then r the coverof C n consisting of its non-trivialcyclic subgroups≥is irredundant of size a 2 2n 1 while C n has an epimorphic image isomorphic to C C so σ(C n)=3. 2 2 2 2 − × We are interested in groups with finite covering number. The following result implies that in order to study the behaviour of the function which assigns to each groupwithfinitecoveringnumberitscoveringnumberitisenoughtoconsiderfinite groups. Theorem 1 (Neumann1954). Let G be an infinite group covered by a finite family of cosets of subgroups of G, and suppose that is irredundant. Then every H H H has finite index in G. ∈H Proof. For a proof see Lemma 4.17 in [Neum]. (cid:3) Indeed,if isaminimalcoverofGthenbyTheorem1 H hasfiniteindex H H∈H in G, hence its normal core N has also finite index and T σ(G/N) =σ(G) σ(G/N), ≤|H| ≤ 1 2 MARTINOGARONZI thus σ(G) = σ(G/N). In other words we are reduced to consider the covering number of the finite group G/N. The solvablegroupswerestudiedbyTomkinson. He provedthefollowingresult. Recall that a “chief factor” of a group G is a minimal normal subgroup H/K of a quotient G/K of G. Theorem 2 (Tomkinson). If G is a finite non-cyclic solvable group then σ(G) = q+1 where q is the order of the smallest chief factor H/K of G with more than one complement in G/K. Note that the number q in the statement of Theorem 2 is a prime power. Not every σ(G) is of the form q+1 with q a prime power,for example σ(Sym(6))=13 (cfr. [S6]). Assume we want to compute the covering number of a group G. If there exists NEGwithσ(G)=σ(G/N)thenwemayconsideraswellthequotientG/N instead of G. This leads instantly to the following definition. Definition 1 (σ-elementary groups). We say that a group G is “σ-elementary” if σ(G)<σ(G/N) for every non-trivial normal subgroup N of G. Clearly,everygrouphasaσ-elementaryquotientwiththesamecoveringnumber. It follows that the structure of the σ-elementary groups is of big interest. It was studied by Lucchini and Detomi in [Spr]. They conjectured that: Conjecture 1. Every non-abelian σ-elementary group is monolithic. Here a groupis said to be “monolithic” if it admits exactly one minimal normal subgroup. 1. Covering nilpotent groups Inthissectionwewillcomputethe coveringnumberofnilpotentgroupsinorder to get the reader familiarized with the methods. Let p be a prime. Observe that the group C C admits exactly p+1 proper p p × subgroups,andallthesesubgroupsarecyclicoforderpandindexp. Letusvisualize this in the subgroup lattice: C C ✇✇✇✇✇✇✇✇✇p× p ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ •●●●●●●●●● • ❧❧❧❧❧❧❧❧❧·❧·❧·❧❧❧❧❧❧• 1 { } Therefore there is a unique cover of C C , it is the one consisting of all of its p p × non-trivial proper subgroups. We obtain that σ(C C )=p+1. p p × The following result (which generalizes the equality σ(C C ) = p+1) is a p p × direct consequence of Theorem 2. However, we will prove it in detail. Proposition 1. Let G be a finite nilpotent group. Then σ(G) = p+1 where p is the smallest prime divisor of G such that the Sylow p-subgroup of G is not cyclic. | | COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS 3 LetusfirstobservethatifGisanyfinitegroupandΦ(G)istheFrattinisubgroup ofG(i.e. theintersectionofthemaximalsubgroupsofG)thenσ(G)=σ(G/Φ(G)). Indeed, in any minimal cover of G consisting of maximal subgroups its members all contain the Frattini subgroup. Now suppose G is a non-cyclic p-group. It is well known that G/Φ(G) = C d ∼ p wheredisthesmallestsizeofasubsetofGgeneratingG. Thereforeσ(G)=σ(C d). p The covering number of C d can be easily computed using the following basic p lemma. Lemma 1 (Minimal Index Lower Bound). Let be a minimal cover of a finite H group T. Then min T :H : H <σ(T). {| | ∈H} Proof. Write = H ,...,H , k = σ(T), β := T : H with β β . 1 k i i 1 k H { } | | ≤ ··· ≤ Since the union H H is not disjoint (because 1 H for i = 1,...,k), we 1 k i ∪···∪ ∈ have k k k T = H < H = T /β k T /β . i i i 1 | | | | | | | | ≤ | | i[=1 Xi=1 Xi=1 It follows that β <k =σ(T). (cid:3) 1 Lemma1impliesthatσ(C d)>p. Ontheotherhand,sinced>1(becauseGis p non-cyclic),C d projectsontoC 2 =C C ,thereforep<σ(C d) σ(C C )= p p p p p p p × ≤ × p+1. We deduce that σ(G) =σ(C d)=p+1. Since any finite nilpotent group is p the direct product of its Sylow subgroups,Proposition1 follows from the following lemma. Lemma 2. Let A,B be two finite groups of coprime order. Then σ(A B)=min σ(A),σ(B) . × { } Proof. Let π :A B A, π :A B B be the canonical projections. Let A B × → × → H be a minimal cover of A B consisting of maximal subgroups, and let × Ω := H : π (H)=B , Ω := H : π (H)=A . A B B A { ∈H } { ∈H } Since A, B arecoprime,anysubgroupofA B isofthe formC D withC A | | | | × × ≤ and D B. It follows that =Ω Ω . Let A B ≤ H ∪ O :=A C, O :=B D. A B − − C×[B∈ΩA A×D[∈ΩB Since covers A B, it coversO O , so O O = . Hence, either O = , A B A B A H × × × ∅ ∅ implying Ω = by minimality of and σ(A B) = σ(A), or O = , implying B B ∅ H × ∅ Ω = by minimality of and σ(A B)=σ(B). (cid:3) A ∅ H × 2. Direct products of groups The very first case to consider when dealing with Conjecture 1 is the direct product case. In a joint work with A. Lucchini we deal with this case. We prove Theorem 3 (Lucchini A., GaronziM.2010[GL]). Let be a minimal cover of a M direct product G=H H of two groups. Then one of the following holds: 1 2 × (1) = X H X where is a minimal cover of H . In this case 2 1 M { × | ∈ X} X σ(G)=σ(H ). 1 4 MARTINOGARONZI (2) = H X X where is a minimal cover of H . In this case 1 2 M { × | ∈ X} X σ(G)=σ(H ). 2 (3) There exist N E H , N E H with H /N = H /N = C and 1 1 2 2 1 1 ∼ 2 2 ∼ p M consists of the maximal subgroups of H H containing N N . In this 1 2 1 2 × × case σ(G)=p+1. Wewillnowgivetheideaofhowthe proofgoeswhenH andH areisomorphic 1 2 non-abelian simple groups. This does not cover all the ideas of the proof but it covers quite well those used when H and H do not have common abelian factor 1 2 groups. Let S be a non-abelian simple group. We want to prove that σ(S S)=σ(S). × Note that since S is a quotient of S S, σ(S S) σ(S). × × ≤ (1) We know that the maximal subgroups of S S are of the following three × types: (1) K S, (2) S K, (3) ∆ := (x,ϕ(x)) x S , ϕ × × { | ∈ } where K is a maximal subgroup of S and ϕ Aut(S). ∈ (2) Let = be a minimal coverof S S, where consists 1 2 3 i M M ∪M ∪M × M of subgroups of type (i). (3) LetΩ:=S S M =Ω Ω ,where Ω =S K × − M∈M1∪M2 1× 2 1 − K×S∈M1 and Ω2 =S− SS×K∈M2K. S (4) We claim thatSit is enough to prove that Ω = . Indeed if this is the ∅ case then either Ω = , in which case K = S and = 1 ∅ K×S∈M1 M M1 by minimality of M, or Ω2 = ∅, in whSich case S×K∈M2K = S, and = 2 by minimality of . In both cases we oSbtain σ(S S) σ(S) M M M × ≥ and hence σ(S S)=σ(S). × Suppose by contradiction Ω= , i.e. Ω = =Ω , and let ω Ω . 1 2 1 6 ∅ 6 ∅6 ∈ (5) The family K <S S K ϕ(ω) ∆ 2 ϕ 3 { | × ∈M }∪{h i | ∈M } is a cover of S of size + (it consists of proper subgroups being 2 3 |M | |M | S non-abelian). Indeed, if b S is such that b K for any K < S such ∈ 6∈ that S K then (ω,b) S S Ω Ω hence, being a cover 2 1 2 × ∈ M ∈ × − × M for S S, (ω,b) ∆ for some ϕ Aut(S) such that ∆ , and we ϕ ϕ 3 × ∈ ∈ ∈ M conclude that b=ϕ(ω) ϕ(ω) . ∈h i (6) It follows that + + = =σ(S S) σ(S) + . 1 2 3 2 3 |M | |M | |M | |M| × ≤ ≤|M | |M | This implies that = . Analogously = . So = . 1 2 3 M ∅ M ∅ M M (7) Observe that since S is coveredby its non-trivial cyclic subgroups, σ(S)< S . Since each member of = has index S , by the Minimal Index 3 | | M M | | Lower Bound (Lemma 1) S <σ(S S) σ(S)< S , | | × ≤ | | a contradiction. COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS 5 3. Sigma star Recall that a groupG is called“primitive” if it admits a core-freemaximal sub- group, that is, a maximal subgroup M such that gMg−1 = 1 . A primitive g∈G { } grouphas always at most two minimal normal subTgroup, and if they are two, they are non-abelian. Recall that a G-group is a group A endowed with a homomorphism f : G → Aut(A). If a A and g G, the element f(g)(a) is usually denoted ag if no ∈ ∈ ambiguity is possible. Definition 2. Let G be a group, and let A,B be two G-groups. A,B are said to be G-isomorphic (written A = B) if there exists an • ∼G isomorphism ϕ:A B such that aϕg =agϕ for every g G. → ∈ A,B are said to be G-equivalent (written A B) if there exist isomor- G • ∼ phisms ϕ:A // B , Φ:G⋉A // G⋉B such that the following diagram commutes: 1 //A //G⋉A // G // 1 { } { } ϕ Φ (cid:15)(cid:15) (cid:15)(cid:15) 1 // B // G⋉B // G // 1 { } { } Let N be a minimal normal subgroup of a group G. The conjugation action of G on N gives N the structure of G-group. Define I (N) to be the set of elements G of G which induce by conjugation an inner automorphism of N and define R (N) G tobe theintersectionofthenormalsubgroupsK ofGcontainedinI (N)withthe G propertythatI (N)/K isnon-Frattini(i.e. notcontainedintheFrattinisubgroup G of G/K) and G-equivalent to N. Recall that the “socle” of a group G, denoted soc(G), is the subgroup of G generatedbytheminimalnormalsubgroupsofG. soc(G)isalwaysadirectproduct of some minimal normal subgroups of G. G is said to be “monolithic” if it admits a unique minimal normal subgroup, i.e. if soc(G) is a minimal normalsubgroup of G. Theorem 4 (Lucchini, Detomi [Spr] Corollary 14). Let H be a non-abelian σ- elementary group and let N ,...,N be minimal normal subgroups of H such that 1 ℓ soc(H) = N N . Let X := G/R (N ) for i = 1,...,ℓ. Then X is a 1 ℓ i H i i ×···× primitive monolithic group with socle isomorphic to N for i = 1,...,ℓ (X will i i be called “the primitive monolithic group associated to N ”) and H is a subdirect i product of X ,...,X : the canonical homomorphism 1 ℓ H X ... X 1 ℓ → × × is injective. Definition 3 (Sigma star). Let X be a primitive monolithic group, and let N be its uniqueminimal normal subgroup. If Ω is an arbitrary union of cosets of N in X define σ (X) to be the smallest number of supplements of N in X needed to cover Ω Ω. If Ω= Nx we will write σ (X) instead of σ (X). Define Nx {Nx} { } σ∗(X):=min σ (X) Ω= Nω , Ω =X . Ω i { | h i } [i 6 MARTINOGARONZI Proposition 2 (Lucchini, Detomi [Spr] Proposition 16). Let H be a non-abelian σ-elementary group with socle N N , 1 ℓ ×···× H subd X1 ... Xℓ ≤ × × as in Theorem 4. For i = 1,...,ℓ let ℓ (N ) be the smallest primitivity degree of Xi i X , i.e. the smallest index of a proper supplement of N in X . Then ℓ (N ) i i i Xi i ≤ σ∗(X ) for i=1,...,ℓ and i ℓ ℓ ℓ (N ) σ∗(X ) σ(H). Xi i ≤ i ≤ Xi=1 Xi=1 Proposition 3 ([Spr], Proposition 10). Let G be a finite group. If V is a comple- mented normal abelian subgroup of G and V Z(G)= 1 then σ(G) 2V 1. ∩ { } ≤ | |− Proof. Let H be a complement of V in G. The idea is to show that G is covered by the family Hv v V C (v)V 1=v V . We omit the details. (cid:3) H { | ∈ }∪{ | 6 ∈ } 4. Small covering numbers The content of this section is included in my Ph.D. thesis. Lemma 3. Let N be a normal subgroup of a group X. If a set of subgroups of X covers a coset yN of N in X, then it also covers every coset yαN with α prime to y . | | Proof. Let s be an integer such that sα 1 mod y . As s is prime to y , by ≡ | | | | Dirichlet’stheoremthereexistinfinitely manyprimesinthe arithmeticprogression s+ y n n N ; we choose a prime p > X in s+ y n n N . Clearly, { | | | ∈ } | | { | | | ∈ } yp =ys. As p is prime to X , there exists an integer r such that pr 1 mod X . | | ≡ | | Hence, if yN M , for every g yαN we have that gp (yα)pN =(yα)sN = i∈I i ⊆∪ ∈ ∈ yN M and therefore also g =(gp)r belongs to M . (cid:3) i∈I i i∈I i ⊆∪ ∪ Proposition 4. Let H be a non-abelian σ-elementary group such that σ(H) 55. ≤ Then H is primitive and monolithic. Proof. We will use the notations of Theorem 4. It is proven in [Spr] that any non-abelian σ-elementary group has at most one abelian minimal normal subgroup. Therefore we may assume that there exists a non-abelian minimal normal subgroup N of H. Let G be the primitive monolithic groupassociatedtoN. IfGhasaprimitivitydegreeatmost27theneitherℓ (N) G ≥ 10andG/N C C ,Sym(3),D (byinspection)-contradictingtheinequality 2 2 8 ∈{ × } ℓ (N) σ(H) σ(G) (being σ(C C ) = σ(D ) = 3 and σ(S ) = 4) - or G/N G 2 2 8 3 ≤ ≤ × is cyclic of prime-power order. Assume the latter case holds. Then G/N admits only one maximal subgroup. In other words, a subset of G generates G modulo N if and only if it contains an element g G such that G/N = gN . Thus Lemma ∈ h i 3 implies that σ(G) σ∗(G)+1, so that ≤ σ∗(X )+σ∗(X ) σ(H) σ(X ) σ∗(X )+1. 1 2 1 1 ≤ ≤ ≤ In particular ℓ (N ) σ∗(X ) 1, and this is a contradiction (ℓ (N ) is the X2 2 ≤ 2 ≤ X2 2 index of a proper subgroup of X ). 2 Thereforewemayassumethatℓ (N) 28wheneverN isanon-abelianminimal G ≥ normal subgroup of G. Suppose H has at least two minimal normal subgroups N =N,N . IfN isnon-abelianthenbyassumptionℓ (N ) 28andProposition 1 2 2 X2 2 ≥ COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS 7 σ Groups 3 C C 2 2 × 4 C C ,Sym(3) 3 3 × 5 Alt(4) 6 C C ,D ,AGL(1,5) 5 5 10 × 7 ∅ 8 C C ,D ,7:3,AGL(1,7) 7 7 14 × 9 AGL(1,8) 10 32 :4,AGL(1,9),Alt(5) 11 ∅ 12 C C ,11:5,D ,AGL(1,11) 11 11 22 × 13 Sym(6) 14 C C ,D ,13:3,13:4,13:6,AGL(1,13) 13 13 26 × 15 SL(3,2) 16 Sym(5),Alt(6) 17 24 :5,AGL(1,16) 18 C C ,D ,17:4,17:8,AGL(1,17) 17 17 34 × 19 ∅ 20 C C ,AGL(1,19),D ,19:3,19:6,19:9 19 19 38 × 21 ∅ 22 ∅ 23 M 11 24 C C ,D ,23:11,AGL(1,23) 23 23 46 × 25 ∅ Table 1. The list of σ-elementary groups G with 3 σ(G) 25. ≤ ≤ 2 implies 56 ℓ (N )+ℓ (N ) σ(H), a contradiction. Hence N is abelian. ≤ X1 1 X2 2 ≤ 2 We have ℓ (N )= N and by Proposition 2 and Proposition 3 X2 2 | 2| 28+ N ℓ (N )+ℓ (N ) σ(H) σ(X )<2N , | 2|≤ X1 1 X2 2 ≤ ≤ 2 | 2| therefore σ(H) 28 N > 1σ(H), and this implies σ(H)>56, a contradiction. − ≥| 2| 2 (cid:3) Proposition4allowsustolisttheσ-elementarygroupswithsmallcoveringnum- ber. Indeed,ifH isaσ-elementarygroupsuchthatσ(H) 55thenH isaprimitive ≤ monolithic group with a primitivity degree at most 55 (cf. Proposition 2). Since thereareonlyfinitely manygroupsofagivenprimitivitydegree,we arereducedto look at a finite list of groups. By giving bounds to their covering numbers we can list the σ-elementary groups G with σ(G) 25. The explicit bounds can be found ≤ in [G25]. In general, the following fact holds. Proposition 5. For every fixed positive integer n, the set of σ-elementary groups H with σ(H)=n is finite, bounded by a function of n. Proof. We will use the notations of Theorem 4. Let H be a σ-elementary group, and write soc(H) = N ... N . Let X ,...,X be the primitive monolithic 1 ℓ 1 ℓ × × groups associated to N ,...,N respectively. H embeds in X ... X , so in 1 ℓ 1 ℓ × × 8 MARTINOGARONZI order to conclude it suffices to bound the number of possibilities for ℓ and each X i in terms of σ(H). By Proposition 2 ℓ ℓ ℓ ℓ (N ) σ∗(X ) σ(H). ≤ Xi i ≤ i ≤ Xi=1 Xi=1 Since there are finitely many primitive groups with a given primitivity degree, the result follows. (cid:3) 5. Considering some monolithic groups The content of this section is included in my Ph.D. thesis. Proposition 4 holds also for 56, but for this number a quite different argument is needed. This is interesting because of the following result, which is [Gmon, Theorem 2]. Here A C denotes the wreath product of A with C , i.e. the 5 2 5 2 semidirect product (A ≀ A )⋊C with the action of C = ε on A A given 5 5 2 2 5 5 × h i × by (x,y)ε =(y,x). Theorem 5 ([Gmon] Theorem 2). σ(A C )=1+4 5+6 6=57. 5 2 ≀ · · AminimalcoverofG=A C is givenbyits socle,soc(G)=A A ,together 5 2 5 5 ≀ × with the subgroups of the form N (M Ma) where a A and M is either G 5 × ∈ the stabilizer of j 1,2,3,4,5 i (for some i 1,2,3,4,5 ) in A or the 5 ∈ { }−{ } ∈ { } normalizer of a Sylow 5-subgroup of A . 5 Thelowerboundsforthecoveringnumberwillbeobtainedbyusingthefollowing tool, introduced by Maro´ti in [MarS]. Definition 4 (Definite unbeatability). Let X be a group. Let be a set of proper H subgroups of X, and let Π X. Suppose that the following four conditions hold for ⊆ and Π. H (1) Π H = for every H ; ∩ 6 ∅ ∈H (2) Π H; ⊆ H∈H (3) Π HS1 H2 = for every distinct pair of subgroups H1 and H2 of ; ∩ ∩ ∅ H (4) Π K Π H for every H and K <X with K . | ∩ |≤| ∩ | ∈H 6∈H Then is said to be definitely unbeatable on Π. H For Π X let σ (Π) be the leastcardinality of a family of proper subgroups of X ⊆ X whose union contains Π. The following lemma is straightforward. Lemma 4. If is definitely unbeatable on Π then σ (Π)= . X H |H| It follows that if is definitely unbeatable on Π then =σ (Π) σ(X). X H |H| ≤ Let us give [MarS, Theorem 3.1] as an example. Let n 11 be an odd integer, ≥ and let X :=Sym(n) be the symmetric group on n letters. Let be the family of H subgroupsofSym(n)consistingofthealternatinggroupAlt(n)andtheintransitive maximal subgroups of Sym(n). Let Π be the subset of Sym(n) consisting of the permutations which are product of at most two disjoint cycles. Then is a cover H of Sym(n) which is definitely unbeatable on Π, therefore σ(Sym(n))= =2n−1. |H| ThisexamplewasrivisitedandgeneralizedbyMaro´tiandme(cf. [MG],[Gmon]) and the results summarized in Theorems 6 and 7 below were obtained. Let us fix some notations we will often use. COVERING MONOLITHIC GROUPS WITH PROPER SUBGROUPS 9 Notations 1. Let G be a monolithic group with socle N = soc(G) = S 1 ×···× S , where S ,...,S are pairwise isomorphic non-abelian simple groups. X := m 1 m N (S )/C (S )isanalmost-simplegroupwithsocleS :=S C (S )/C (S )=S . G 1 G 1 1 G 1 G 1 ∼ 1 The minimal normal subgroups of Sm = S ... S are precisely its factors, 1 m × × S ,...,S . Since automorphisms send minimal normal subgroups to minimal nor- 1 m malsubgroups,itfollows thatGactsonthemfactors ofN. Letρ:G Sym(m)be → thehomomorphism induced by the conjugation action of G on the set S ,...,S . 1 m { } K := ρ(G) is a transitive permutation group of degree m. By [BEcl, Remark 1.1.40.13] G embeds in the wreath product X K. Let L be the subgroup of X ≀ generated by the following set: S x x k K : (x ,...,x )k G . 1 m 1 m ∪{ ··· | ∃ ∈ ∈ } LetT bea normal subgroupof X containingS and contained in L with theproperty that L/T has prime order if L=S, and T =L if L=S. 6 Let G be a primitive monolithic group with non-abelian socle N, and write N = Sm with S a non-abelian simple group. The covers of G we often look at consist of some subgroups of G containing N and subgroups of the form N (M G × Ma2 Mam) with M <S, which will be called “product type subgroups”. ×···× Inthe followingifn isa positiveintegerwedenote byω(n)the number ofprime divisors of n. Suppose that G/N is cyclic. The covers of G we consider consist of all the ω(G/N ) maximal subgroups of G containing N and some product type | | subgroupsNG((S M) (S M)a2 (S M)am)wherea1 =1,a2,...,am S ∩ × ∩ ×···× ∩ ∈ andM variesinafamilyofmaximalsubgroupsofX supplementingS whichcovers a cosetxS of S in X which generatesthe cyclic groupX/S. This is how we obtain upper bounds for σ(G) (the size of a cover of G is an upper bound for σ(G)). Theorem 6 (Mar´otiA.,GaronziM.2010[MG]). Let G be a monolithic group with non-abelian socle, and let us use Notations 1. Suppose that G/N is cyclic and that X =S =Alt(n). Then the following holds. (1) If 12<n 2 mod (4) then ≡ (n/2)−2 n m 1 n m σ(G)=ω(m)+ + . (cid:18)i(cid:19) 2m(cid:18)n/2(cid:19) i=1X, i odd (2) If 12<n 2 mod (4) then 6≡ n m 1 n ω(m)+ σ(G). 2 (cid:18)i(cid:19) ≤ i=1X, i odd (3) Suppose n has a prime divisor at most √3n. Then σ(G) ω(m)+min S :M m−1 as n . ∼ M MX∈M| | →∞ Theorem 7 (Garonzi M. 2011 [Gmon]). Let G be a monolithic group with non- abelian socle, and let us use Notations 1. Suppose that G/N is cyclic and that X =Sym(n). Then the following holds. (1) Suppose that n 7 is odd and (n,m)=(9,1). Then ≥ 6 (n−1)/2 m n σ(G)=ω(2m)+ . (cid:18)i(cid:19) Xi=1 10 MARTINOGARONZI (2) Suppose that n 8 is even. Then ≥ m m [n/3] m 1 n 1 n n σ(G) ω(2m)+ + . (cid:18)2(cid:18)n/2(cid:19)(cid:19) ≤ ≤ (cid:18)2(cid:18)n/2(cid:19)(cid:19) (cid:18)i(cid:19) Xi=1 m In particular σ(G) 1 n as n . ∼(cid:16)2(cid:0)n/2(cid:1)(cid:17) →∞ 6. Attacking the conjecture The content of this section is included in my Ph.D. thesis. The following result provides a first partial reduction to monolithic groups. Proposition6. LetH beanon-abelianσ-elementarygroup,letN ,...,N bemin- 1 ℓ imal normal subgroups of H such that soc(H)=N N andlet X ,...,X be 1 ℓ 1 ℓ ×···× the primitive monolithic groups associated to N ,...,N respectively. Then at most 1 ℓ one of N ,...,N is abelian. Suppose that N is non-abelian and that σ∗(X ) 1 ℓ 1 1 ≤ σ∗(X ) whenever j 1,...,ℓ and N is non-abelian. If σ(X ) < 2σ∗(X ) then j j 1 1 ∈ { } H =X , i.e. H is monolithic. ∼ 1 Proof. By Proposition 2 ℓ σ∗(X )+ σ∗(X ) σ(H) σ(X )<2σ∗(X ). 1 j 1 1 ≤ ≤ Xj=2 It follows that ℓ σ∗(X )<σ∗(X ) hence, by the minimality hypothesis on X , j=2 j 1 1 N2,...,Nℓ arePabelian. In [Spr, Corollary 14] it is proved that any non-abelian σ-elementarygrouphas at mostone abelianminimal normalsubgroup,thus ℓ=2. Since N is abelian ℓ (N )= N , and by Proposition 2 2 X2 2 | 2| min 2σ∗(X ),2N σ∗(X )+ N =σ∗(X )+ℓ (N ) { 1 | 2|} ≤ 1 | 2| 1 X2 2 ≤ σ(H) min σ(X ),σ(X ) . 1 2 ≤ ≤ { } Now by hypothesis σ(X ) < 2σ∗(X ), and σ(X ) < 2N by Proposition 3. This 1 1 2 2 | | leads to a contradiction. (cid:3) In order to provean inequality like σ(G)<2σ∗(G) for G a primitive monolithic group we first need some way to get as much general as possible upper bounds for σ(G). Theorem 8. Let G be a monolithic group with non-abelian socle, and let us use Notations 1. Assume that X/S is abelian. Let be a set of maximal subgroups M of X supplementing S and such that M contains a coset xS L with the M∈M ∈ property that x,T =L. S h i Then σ(G) 2m−1+ S :S M m−1. ≤ M∈M| ∩ | P Unfortunately the hypothesis “X/S abelian” does not seem easy to bypass. Proof. If L=T define 6 R:= (x ,...,x )k G x x T . 1 m 1 m { ∈ | ··· ∈ } Since X/S is abelian, R is a proper subgroup of G.