Advances in Group Theory and Applications c 2020 AGTA - www.advgrouptheory.com/journal 9 (2020), pp. 39–49 ISSN: 2499-1287 DOI: 10.32037/agta-2020-002
Generation of Finite Groups and Maximal Subgroup Growth
Andrea Lucchini —Mariapia Moscatiello
(Received May 8, 2019;AcceptedJune22, 2019 —CommunicatedbyF.deGiovanni)
Abstract
Let G be a finite group and, for n N, denote by mn(G) the number of maximal subgroups of G with index n.Let 2
M(G)=sup log mn(G)/log n n>2 and let E1(G) be the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found. Then
M(G) - 4 E (G) M(G) + 3. d e 6 1 6 d e
Mathematics Subject Classification (2010): 20P05 Keywords:waitingtime;generationoffinitegroups
1 Introduction
Let G be a nontrivial finite group and let x =(xn)n N be a sequence of independent, uniformly distributed G-valued random2 variables. We may define a random variable ⌧G (a waiting time) by
⌧ = min{n 1 | x , ..., xn = G} [1, + ]. G > h 1 i 2 1 40 A. Lucchini – M. Moscatiello
Notice that ⌧ >nif and only if x , ..., xn = G,sowehave G h 1 i6
P(⌧G >n)=1 - PG(n), denoting by
n |{(g , ..., gn) G | g , ..., gn = G}| P (n)= 1 2 h 1 i G |G|n the probability that n randomly chosen elements of G generate G. We denote by E1(G) the expectation E(⌧G) of this random variable. In other word E1(G) is the expected number of elements of G which have to be drawn at random, with replacement, before a set of gener- ators is found.
Significant estimations for the value of E1(G) have been obtained by A. Lubotzky in [5]. More precisely he estimated another related invariant, defined by I. Pak:
1 V(G)=min k N P (k) . 2 G > e As it was noticed by I. Pak (see for example Proposition 1.1 of [5]), E1(G) and V(G) are related in the following way:
1 e E (G) V(G) E (G).(1.1) e · 1 6 6 e - 1 · 1
For n N,denotebymn(G) the number of maximal subgroups of G 2 with index n and let
log mn(G) M(G)=sup . log n n>2
Actually M(G) is the “polynomial degree” of the rate of growth of mn(G).Thisratehasbeenstudiedforfiniteandprofinitegroups by Mann, Shalev, Borovik, Jaikin-Zapirain, Liebeck, Pyber and more recently by Ballester-Bolinches, Esteban-Romero, Jiménez-Seral and Hangyang Meng (see [2],[3],[9],[10],[1]). It is roughly equal to V(G), indeed we have (see Proposition 1.2 of [5]):
M(G)-3.5 6 V(G) 6 M(G)+2.02.(1.2) Generation of finite groups and maximal subgroup growth 41
The estimation for E1(G) given by Lubotzky is obtained combin- ing (1.1)and(1.2):
e - 1 (M(G)-3.5) E (G) (M(G)+2.02) e.(1.3) · e 6 1 6 · However this estimation leaves open the question whether
|M(G)-E1(G)| could be arbitrarily large. The main observation of this paper is that the arguments used in [5]canbeimprovedandthefollowingresult can be obtained.
Theorem 1.1 Let G be a finite group. Then
M(G) - 4 E (G) M(G) + 3. d e 6 1 6 d e
Other numerical invariants may be derived from ⌧G starting from the higher moments
k k E(⌧G)= n P(⌧G = n). nX>1 In particular it is probabilistically important, when the expectation of arandomvariableisknown,tohavecontroloveritssecondmoment. 2 We will denote by E2(G) the second moment E(⌧G) and by
2 var(⌧G)=E2(G)-E1(G) the variance of ⌧G.
Theorem 1.2 Let G be a finite group. Then
⇡2 M(G) 2 - 8 M(G) + 13 E (G) M(G) 2 + 6 M(G) + 14 + . d e d e 6 2 6 d e d e 3
2 Corollary 1.3 var(⌧ ) 14 M(G) + ⇡ - 2. G 6 d e 3
In [7]itisprovedthatE1(G) and E2(G) can be directly determined using the Möbius function defined on the subgroup lattice of G by setting µ(G)=1 and µ(H)=- H µ(H)|G| µ(H)|G|(|G| + |H|) E (G)=- , E (G)=- (1.4) 1 |G| - |H| 2 (|G| - |H|)2 H µ(H)|G| M(G) - 4 - M(G) + 3, d e 6 |G| - |H| 6 d e H The definition of E1(G) can be extended to the case of a (topolog- ically) finitely generated profinite group G.Ifwedenotewithµ the normalized Haar measure on G,sothatµ(G)=1,theprobability that k random elements generate (topologically) G is defined as P (k)=µ({(x , ..., x ) Gk| x , ..., x = G}), G 1 k 2 h 1 ki where µ denotes also the product measure on Gk.Asitisprovedfor example in Section 6 of [7], E1(G)= sup E1(G/N), N N 2 where N is the set of the open normal subgroups of G.Thisimplies that the inequalities in Theorems 1.1 and 1.2 still hold if G is a finitely generated profinite group. Recall that a profinite group G is said to Generation of finite groups and maximal subgroup growth 43 be positively finitely generated, PFG for short, if PG(k) is positive for some natural number k and that G is said to have polynomial maximal subgroup growth if mn(G) 6 ↵n for all n (and for some constant ↵ and ). The profinite version of Theorem 1.1 can be con- sidered as a quantitative version of the celebrated result of Mann and Shalev [10], saying that a profinite group is PFG if and only if it has polynomial maximal subgroup growth. 2 Proof of Theorem 1.1 For a real number ⌘ > 1,letusdefine 1 V⌘(G)=min k N P (k) . 2 G > ⌘ The argument used by Lubotzky to bound V(G)=Ve(G),canbe adapted to bound V⌘(G),foranarbitrarilyvalueof⌘.Theproofis essentially the same, but for reader’s convenience we prefer to give the details. In the following, when we write log we always take it in base 2. It was proved by Pyber (see Theorem 1.3 of [5]) that there exists a constant b such that for every finite group G and every n > 2, G has at most nb core-free maximal subgroups of index n.Infact,b = 2 will do. Lemma 2.1 V⌘(G) > M(G)-b - log ⌘ > M(G)-2 - log ⌘. Proof —LetNi be an enumeration of all cores of maximal sub- groups of G (each core occuring only once). For each Ni choose amaximalsubgroupMi whose core is Ni.LetCn(G) be the num- ber of the maximal subgroups of index n obtained in this way. The k k events Mi in G are pairwise independent and from the quantitative version of the Borel-Cantelli lemma, one deduces that -k 1 Cn(G)n 6 1 Pk(G) nX=2 and in particular, nk Cn(G) 6 . Pk(G) 44 A. Lucchini – M. Moscatiello Taking k = V⌘(G) we get that V⌘(G) Cn(G) ⌘ n . 6 · Now, Pyber’s Theorem implies that b mn(G) 6 Cn(G)n . V (G)+b Hence, mn(G) ⌘ n ⌘ .Itfollowsthat 6 · log mn(G) M(G)=sup 6 V⌘(G)+b + log ⌘. log n n>2 This concludes our proof. ut We recall the following formula (see for example (1.1)of[7]). Lemma 2.2 E (G)= (1 - P (n)). 1 n>0 G Lemma 2.3 M(G) -P4 E (G). d e 6 1 Proof —ByLemma2.1,foreverypositiveintegeri,wehave i V2i (G) > M(G)-b - log(2 ) > M(G)-2 - i. In particular if k = M(G) - 3 - i,thenP (k) <2-i.Letm = M(G) . d e G d e It follows from Lemma 2.2 that k-(m-3) E1(G) > (1 - PG(k)) > 1 - 2 06kX6m-4 06kX6m-4 ⇣ ⌘ -j -j = m - 3 - 2 > m - 3 - 2 = m - 4. 16jX6m-3 jX>1 This concludes our proof. ut Lemma 2.4 E (G) M(G) - 3. 1 6 d e Proof —Letm = M(G) and k = m + t with t apositiveinteger. d e As it is noticed in the proof of Proposition 11.2.2 of [6]wehave M(G) m mn(G) n n 1 1 - P (k) . G 6 nk 6 nk 6 nk 6 nt nX>2 nX>2 nX>2 nX>2 Generation of finite groups and maximal subgroup growth 45 It follows that E1(G)= (1 - PG(k)) 6 m + 2 + (1 - PG(k)) kX>0 k>Xm+2 m + 2 + n-u = m + 2 + n-u 6 0 1 0 0 11 uX>2 nX>2 nX>2 uX>2 @ n A @ 1@ AA = m + 2 + = m + 2 + = m + 3. n2(n - 1) 0 n(n + 1)1 nX>2 nX>1 @ A This concludes our proof. ut 3 Proof of Theorem 1.2 We recall the following formula (see for example Lemma 18 of [7]). Lemma 3.1 E (G)+E (G)=2 (n + 1)(1 - P (n)). 1 2 n>0 G P 2 Lemma 3.2 E (G)+E (G) M(G) 2 + 7 M(G) + 10 + ⇡ . 1 2 6 d e d e 3 Proof —Settingm = M(G) and using Lemma 3.1,wededuce d e E1(G)+E2(G)=2 (k + 1)(1 - PG(k)) kX>0 6 2 (k + 1)+2 (k + 1)(1 - PG(k)) 06k 2 1 n(m + 2) = + 0 nu 1 n2(n - 1) nX>2 uX>1 nX>2 @ A 1 2 ⇡2 = + m + 2 = m + 2 + . n - 1 6 nX>2 ✓ ◆ This concludes our proof. ut 2 Corollary 3.3 E (G) M(G) 2 + 6 M(G) + 14 + ⇡ . 2 6 d e d e 3 Proof —CombiningLemma3.2 and Lemma 2.3,wededuce ⇡2 E (G) M(G) 2 + 7 M(G) + 10 + - E (G) 2 6 d e d e 3 1 ⇡2 M(G) 2 + 7 M(G) + 10 + -( M(G) - 4) 6 d e d e 3 d e ⇡2 = M(G) 2 + 6 M(G) + 14 + . d e d e 3 This concludes our proof. ut Lemma 3.4 E (G)+E (G) M(G) 2 - 7 M(G) + 10. 1 2 > d e d e Proof —Settingm = M(G) and applying Lemmas 2.1 and 3.1, d e we obtain e (G)+e (G) 1 2 (k + 1)(1 - P (k)) 2 > G 06kX6m-4 k-(m-3) > (k + 1) 1 - 2 06kX6m-4 ⇣ ⌘ (m - 2)(m - 3) m - 2 - j = - 2 2j 16jX6m-3 (m - 2)(m - 3) j m - 2 + - > 2 2j 2j 16jX6m-3 jX>1 (m - 2)(m - 3) -(m - 2). > 2 This concludes our proof. ut Generation of finite groups and maximal subgroup growth 47 Corollary 3.5 E (G) M(G) 2 - 8 M(G) + 13. 2 > d e d e Proof —CombiningLemma3.4 and Lemma 2.4,wededuce E (G) M(G) 2 - 7 M(G) + 10 - E (G) 2 > d e d e 1 M(G) 2 - 7 M(G) + 10 -( M(G) - 3) > d e d e d e = M(G) 2 - 8 M(G) + 13. d e d e This concludes our proof. ut 4 Proof of Theorem 1.5 It can be easily seen that for every finite soluble group G and every n > 2, G has at most n core-free maximal subgroups of index n. This means that the constant b in the statement of Lemma 2.1 can be taken equal to 1 in the case of finite soluble groups and consequently we have: Lemma 4.1 If G is a finite soluble group, then M(G) - 3 E (G) M(G) + 3. d e 6 1 6 d e Lemma 4.2 Let G be a finite soluble group. There exists a prime divisor p of the order of G and a positive integer a such that a M(G) mpa (G)=p · . If H is a subgroup of G containing a Sylow p-subgroup of G,then M(G) 6 M(H)+3. Proof —Letm = M(G).SincethemaximalsubgroupsofG have prime-power indices, there exists a prime divisor p of the order of G a m and a positive integer a such that mpa (G)=p · .LetH be a sub- group of G containing a Sylow p-subgroup of G and let µ = M(H).It follows from (2.4)of[8]that a m a+b a+b b µ 3 a+µ a p · = mpa (G) 6 p mpb (H) 6 p p · 6 p · · , bX6a bX6a 48 A. Lucchini – M. Moscatiello hence m 3 + µ. 6 ut Proof of Theorem 1.5 —ByLemma4.2,thereexistsaprimedivi- sor p of G,suchthat M(G) 6 M(Gp)+3. But then we deduce from Lemma 4.1,that E (G) M(G) + 3 M(Gp) + 6 E (Gp)+9. 1 6 d e 6 d e 6 1 The theorem is proved. ut REFERENCES [1] A. Ballester-Bolinches – R. Esteban-Romero –P.Jiménez- Seral –Hangyang Meng:“Boundsonthenumberofmaxi- mal subgroups with applications to random generation of finite groups”, preprint. [2] A.V. Borovik – L. Pyber – A. Shalev:“Maximalsubgroupsin finite and profinite groups”, Trans. Amer. Math. Soc. 348 (1996), 3745–3761. [3] A. Jaikin-Zapirain – L. Pyber:“Randomgenerationoffinite and profinite groups and group enumeration”, Ann. Math. 173 (2011), 769–814. [4] L. G Kovács –Hyo-Seob Sim:“Generatingfinitesolublegroups”, Indag. Math. (N.S.) 2 (1991), 229–232. [5] A. Lubotzky:“Theexpectednumberofrandomelementstogen- erate a finite group”, J. Algebra 257 (2002), 452–459. [6] A. Lubotzky – D. Segal:“SubgroupGrowth”,Birkhäuser,Basel (2003). [7] A. Lucchini:“Theexpectednumberofrandomelementstogen- erate a finite group”, Monatsh. Math. 181 (2016), 123-–142. Generation of finite groups and maximal subgroup growth 49 [8] A. Lucchini – M. Moscatiello:“Aprobabilisticversionofa theorem of Laszlo Kovacs and Hyo-Seob Sim”, Int. J. Group The- ory 9 (2020), n.1, 1–6. [9] A. Mann:“Positivelyfinitelygeneratedgroups”,Forum Math. 8 (1996), 429–459. [10] A. Mann – A. Shalev:“Simplegroups,maximalsubgroups and probabilistic aspects of profinite groups”, Israel J. Math. 96 (1996), 449–468. Andrea Lucchini, Mariapia Moscatiello Dipartimento di Matematica “Tullio Levi-Civita” Università di Padova Via Trieste, 63 35131 Padova (Italy) e-mail: [email protected] [email protected]