A note on invariable generation of nonsolvable permutation groups
AA NOTE ON INVARIABLE GENERATION OF NONSOLVABLEPERMUTATION GROUPS
JOACHIM K ¨ONIG AND GICHEOL SHIN
Abstract.
We prove a result on the asymptotic proportion of randomly chosen pairs ( σ, τ ) ofpermutations in the symmetric group S n which “invariably” generate a nonsolvable subgroup,i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of S n .As an application, we obtain that for a large degree “random” integer polynomial f , reductionmodulo two different primes can be expected to suffice to prove the nonsolvability of Gal( f/ Q ). Introduction and main result
This paper is motivated by the following, at this point deliberately vaguely worded question:
Question . Given a “random” integer polynomial f ∈ Z [ X ], how “difficult” is it to verify thenonsolvability of the Galois group Gal( f / Q )?To begin with, the precise meaning of “random polynomial” is up to the choice of a concretemodel. For our purposes, we essentially mean “a degree n polynomial whose Galois group is A n or S n ” (although of course Question 1.1 is to be understood such that whoever aims at provingnonsolvability is provided with no prior information on the Galois group of the concrete polyno-mial), or slightly more general (and in order to allow a notion of randomness) “a polynomial f chosen from a family in such a way that Gal( f / Q ) is almost certainly alternating or symmetric”.The latter is known to hold for several versions of random polynomials. For example, a famousresult by van der Waerden ([16]) asserts that for any fixed degree n ∈ N and for N → ∞ , theproportion of polynomials with Galois group S n among degree- n integer polynomials with coef-ficients in {− N, . . . , N } tends to 1. Moreover, [1] shows that if N is the product of four distinctprimes, then for n → ∞ , the probability of a degree- n polynomial with coefficients chosen uni-formly at random from the set { , . . . , N } to have Galois group containing A n tends to 1. Thisviewpoint (i.e., degree tending to infinity) will be more relevant for us in view of Theorem 1.5below.Next, regarding the difficulty of verification mentioned in Question 1.1, we only consider oneapproach to obtain information about the Galois group of an integer polynomial, based on thefollowing well-known criterion on modulo- p reduction due to Dedekind. Theorem 1.2 (Dedekind) . Let f ∈ Z [ X ] be a separable polynomial of degree n with Galoisgroup G = Gal ( f / Q ) ≤ S n , let and p be a prime dividing neither the leading coefficient of f nor the discriminant of f . Then the Galois group of the mod- p reduction f of f (acting on the Mathematics Subject Classification.
Primary 05A05, 20B35; Secondary 11R32.
Key words and phrases.
Combinatorics; random permutations; permutation groups; solvable groups; Galoisgroups. While Dedekind’s criterion of course in general cannot answer all questions about the Galois group, it isconsiderably less expensive than other approaches such as resolvent methods. a r X i v : . [ m a t h . C O ] F e b JOACHIM K ¨ONIG AND GICHEOL SHIN roots of f ) embeds as a permutation group into G . In particular, if d , . . . , d r are the degrees ofthe irreducible factors of f over F p , then G contains an element whose cycle lengths are exactly d , . . . , d r . Conversely, given f ∈ Z [ X ] with Galois group G ≤ S n , the existence of primes whose reductionyields any prescribed cycle structure inside G is guaranteed and quantified by Frobenius’ densitytheorem (later famously strengthened by Chebotarev). Theorem 1.3 (Frobenius) . Let f ∈ Z [ X ] be a separable degree- n polynomial with Galois group G ≤ S n , and let C := [ c , . . . , c r ] be a cycle type in G . Then the asymptotic density of primes p for which mod- p reduction of f in Theorem 1.2 yields cycle type C equals the proportion ofelements of cycle type C in G . As (a special case of) a well-known theorem of Jordan, a subgroup intersecting all conjugacyclasses of S n nontrivially must be S n itself. Thus, Dedekind’s and Frobenius’ theorems guaranteethat, in the case Gal( f / Q ) = S n , the above reduction process will eventually correctly identifythe Galois group, if sufficiently many primes are chosen. Efficient bounds on the number ofprimes required to succeed with certainty are hard to obtain (and depend of course on theconcrete polynomial f ), see, e.g., [10]. We instead ask how many primes are needed to succeed“with high probability”. Here, surprisingly strong results are available, as outlined below.The following definition is directly motivated by Frobenius’ theorem, which yields, for eachmod- p reduction of a given polynomial f , not a concrete element, but only a cycle type which isguaranteed to occur in the Galois group of f . Definition 1.4.
Let G ≤ S n , let S be a family of subgroups of S n , and σ , . . . , σ r ∈ G . Say that σ , . . . , σ r invariably generate a member of S , if for all τ , . . . , τ r ∈ S n such that τ i is conjugatein S n to σ i for all i = 1 , . . . , r , the group generated by τ , . . . , τ r belongs to S .The dependency of the underlined parts in Definition 1.4 on S n rather than G is deliberate. Ofcourse, switching them from S n to G would also be meaningful, and would indeed be common ina purely group theoretical context. E.g., with this notion of “invariable generation”, it is knownthat every non-abelian simple group G has a pair of elements all of whose G -conjugates generate G , see [9, Theorem 1.3], and that the expected number of G -conjugacy classes (of an arbitrarygroup G ) required to invariably generate G is in O ( (cid:112) | G | ), see [11]. However, our version seemsnatural in view of the Galois theoretical interpretation, since it corresponds to a situation whereno prior information on the true Galois group G ≤ S n is available.For the key case G = S n , invariable generation of transitive subgroups of S n (i.e., S equal tothe set of all transitive subgroups of S n ), or indeed of S n itself, has been studied by a numberof authors. For example, Dixon conjectured ([4]) that an absolutely bounded number of permu-tations in S n should suffice to invariably generate a transitive subgroup of S n with probabilitytending to 1 as n → ∞ . It was later shown that in fact four permutations σ , . . . , σ ∈ S n , cho-sen independently and uniformly at random from the set S n (for short: random permutations),invariably generate a transitive subgroup, and even one of A n or S n (i.e., S = { A n , S n } ), withprobability tending to 1 as n → ∞ , cf. [1], building on the earlier [14]. On the other hand, forthree random permutations, the probability to invariably generate a transitive subgroup tendsto 0 as n → ∞ ([6]). In the context of Dedekind’s criterion, this implies that in order to finda certificate for the irreducibility (or in fact for the assertion Gal( f / Q ) ⊇ A n ) of a large de-gree polynomial with Galois group S n , one should expect to require modulo- p reduction for fourdifferent primes p . NOTE ON INVARIABLE GENERATION OF NONSOLVABLE PERMUTATION GROUPS 3
Motivated by Question 1.1, we are interested in the case where S is the set of all nonsolvablesubgroups of S n . Concretely, we prove the following. Theorem 1.5.
Let G = S n or G = A n . Then the proportion of pairs ( σ, τ ) ∈ G whichinvariably generate a nonsolvable subgroup of S n is − O (1 / (log n ) − (cid:15) ) , for every (cid:15) > . Inparticular, two random permutations σ, τ ∈ G invariably generate a nonsolvable subgroup withprobability tending to as n → ∞ . Regarding Question 1.1, this implies that in order to find a certificate for the nonsolvabilityof Gal( f / Q ) for a given large degree integer polynomial f with Galois group S n or A n , it almostcertainly suffices to reduce modulo two different primes. Obviously this is best possible, sincea single mod- p reduction will only yield one particular cyclic subgroup of Gal( f / Q ), and thuscannot even rule out this group to be cyclic.The proof of Theorem 1.5, contained in Section 3, makes decisive use of a recent combina-torial result by Unger ([15]); in terms of permutation group theory, it requires only elementaryprerequisites. In the appendix, we show how to obtain a more concise version of the nonsolvabil-ity conclusion of Theorem 1.5, which requires some more advanced classification results aboutprimitive groups. 2. Prerequisites
We include some basic combinatorial and group-theoretical facts about permutations whichwill be used later.2.1.
Permutations with partially prescribed cycle structure.
Let λ be a partition of k .For any permutation σ ∈ S k of cycle type λ , we denote by λ ! the size of the centralizer C S k ( σ )in S k , i.e., λ ! = (cid:81) ∞ i =1 i m i ( λ ) m i ( λ )!, where m i ( λ ) denotes the multiplicity of i in λ . Recall thatthe number of permutations of the cycle type corresponding to λ is equal to k ! λ ! .Let λ be a partition of k for some k ≤ n . We denote by N λ,n the number of permutations in S n which contain at least m ( λ ) 1-cycles, m ( λ ) 2-cycles, m ( λ ) 3-cycles, and so on, i.e., N λ,n = { σ ∈ S n : the multiplicity of i in the cycle type of σ is ≥ m i ( λ ) for each i } . To compute (an upper bound of) the number N λ,n , we can pick a k -subset X of { , , . . . , n } in (cid:0) nk (cid:1) ways, then pick a permutation on X of cycle type λ in k ! λ ! ways, and then pick any permutationon { , , . . . , n } \ X in ( n − k )! ways. Thus, the total number of ways is equal to (cid:18) nk (cid:19) · k ! λ ! · ( n − k )! = n ! λ ! . In the above we have counted permutations of cycle type µ exactly (cid:81) ∞ i =1 (cid:0) m i ( µ ) m i ( λ ) (cid:1) times, where µ is a partition of n such that m i ( µ ) ≥ m i ( λ ) for each i . Hence, we have N λ,n ≤ n ! λ ! . Consequently,we have: Lemma 2.1.
Let m , . . . , m n be non-negative integers. The probability that a random permu-tation in S n contains at least m m m (cid:81) ni =1 i m i m i ! . In particular, the probability that a random permutation in S n contains at leastone k -cycle is less than or equal to k . JOACHIM K ¨ONIG AND GICHEOL SHIN
Primitive and imprimitive permutation groups.
A transitive permutation group G ≤ S n is called primitive if it does not preserve a non-trivial block system in { , . . . , n } , or equiv-alently, if the point stabilizers are maximal subgroups in G . The following are classical resultsabout the structure of primitive permutation groups. The first one is due to Jordan (see [5,Theorem 3.3E]), the second one was essentially known to Galois (cf. [13]). Theorem 2.2 (Jordan) . Let G be a primitive permutation group of degree n , containing a cycleof prime length fixing at least three points. Then G ≥ A n . Theorem 2.3.
Let G ≤ S n be a primitive and solvable group. Then G acts as an affine groupon some vector space. In particular, n is a prime power. If furthermore n = p is a prime, then G is contained in the normalizer N S p ( (cid:104) σ (cid:105) ) ∼ = C p (cid:111) C p − =: AGL ( p ) , for some p -cycle σ . Recall furthermore that if G ≤ S n acts transitively but imprimitively, then there exist a, b > n = ab and transitive groups U ≤ S a , V ≤ S b such that G embeds into the wreathproduct U (cid:111) V := U b (cid:111) V (with V acting via permuting the b copies of U ). Furthermore, if G < G denotes a point stabilizer, then there exists G < H < G such that the image of G inthe action on cosets of H equals V , and the image of H in the action on cosets of G in H equals U . See, e.g., [5] for more extensive background on primitive and imprimitive permutationgroups. 3. Proof of Theorem 1.5
Proof.
Firstly, it obviously suffices to consider G = S n . Since A n comprises half the permutationsof S n , the assertion for A n follows readily. We pick two permutations σ, τ ∈ S n independently atrandom. We will show that the following property of σ, τ holds with probability 1 − O (( log log n log n ) ).(1) For all x ∈ S n , the subgroup (cid:104) σ, τ x (cid:105) ≤ S n is nonsolvable . We will achieve this through a series of claims. Note that whenever we encounter a propertyfulfilled by a proportion of O (( log log n log n ) ) pairs of permutations ( σ, τ ) ∈ S n (or in particular, aproperty fulfilled by a proportion of O (( log log n log n ) ) permutations σ ∈ S n ), we may exclude thepermutations fulfilling this property from the further considerations, simply since the union of abounded number of sets of a certain asymptotic O ( f ( n )) is still in O ( f ( n )). Step I : We first claim that the following property holds for a proportion 1 − O (( log log n log n ) ) ofall pairs of permutations ( σ, τ ) ∈ S n :There exists a prime p ∈ [log( n ) , n/
2] such that at least one of σ, τ contains a p − cycle and has all of its other cycle lengths coprime to p. (2)Indeed, it suffices to show that a proportion 1 − O ( log log n log n ) of permutations σ ∈ S n fulfillthe condition in (2). Without the upper bound condition p ≤ n/
2, this is shown in Theorems 2and 5 of [15]. To see that the upper bound may be added without changing the estimate on theasymptotic proportion, note that due to Lemma 2.1, the probability for σ to contain a p -cyclefor some prime p larger than n/ (cid:80) p ∈ P n/
Since log( x ) − log( x − c ) is of the order of growth of cx (for x → ∞ and c constant), we obtainin total that the probability that σ contains a p -cycle for some prime p > n is in O ( n ), anda fortiori in O ( log log n log n ). Thus, the requirement p ≤ n/ σ and τ , andfor some prime p ∈ [log( n ) , n/ σ fulfills the condition, and will furthermore denote the p -cycle of σ by (1 , , . . . , p ), which is ofcourse also without loss of generality. Step II : We next claim that the proportion of σ ∈ S n containing a cycle of length a Mersenneprime ≥ log( n ) is O ( n ) ), whence we may and will additionally assume in the following thatthe prime p in Condition (2) is not a Mersenne prime.Indeed, we may again use Lemma 2.1 together with a simple geometric series estimate, tobound the probability for σ to contain a cycle of some length 2 k − ≥ log( n ) from above by ∞ (cid:80) j =0 1log( n ) · ( ) j , which is in O ( n ) ).We now include the second permutation τ in our considerations, and (for all possible choicesof x ∈ S n ) look at the length of the orbit of the subgroup (cid:104) σ, τ x (cid:105) containing the orbit { , . . . , p } of (cid:104) σ (cid:105) . We denote this orbit by O and from hereon consider the image U of (cid:104) σ, τ x (cid:105) in its actionon O . Of course it suffices to show that for a proportion 1 − O (( log log n log n ) ) of pairs ( σ, τ ), thereexists no x such that U is solvable. Step III : We claim that, for any σ as above, the probability (in τ ) that there exists at leastone x ∈ S n for which the above U ≤ Sym ( O ) is solvable and primitive is O ( n ) ).Indeed, due to Step I, U contains a p -cycle (namely, a suitable power of σ , restricted to O ).Thus, from Theorem 2.2, if U is primitive and solvable, then certainly | O | ∈ { p, p + 1 , p + 2 } . Thecase | O | = p +2 can also be excluded by elementary means. Indeed, by Theorem 2.3, a primitivesolvable group is affine, i.e., contained in
AGL ( V ) ∼ = V (cid:111) GL ( V ) for some finite vector space V . The point stabilizers in AGL ( V ) are conjugate to (the point stabilizer of the zero vector) GL ( V ), so if U ≤ AGL ( V ) contains a cycle fixing exactly two points, then in particular GL ( V )contains an element fixing exactly one non-zero vector. But the latter can only happen if V isan F -vector space, i.e., | O | = p + 2 is a 2-power, which is impossible. Furthermore, the case | O | = p + 1 can be excluded via Step II, since if U were primitive and solvable (and thus, affine)of degree p + 1, then p + 1 would be a prime power, i.e., p would be a Mersenne prime. So weare left with the case that U is primitive and solvable of degree p , i.e., U ≤ AGL ( p ). But theonly elements of AGL ( p ) are p -cycles and powers of a ( p − , d, . . . , d ] for d dividing p −
1. We need to bound the proportion of τ ∈ S n having such a patternin their cycle structure. Of course, since p ≥ log( n ) , the probability for τ to contain a p -cycleis ≤ n ) by Lemma 2.1. On the other hand, via setting a := p − d , the combined probabilityto contain a cycles of length d for any d | p − (cid:80) a | p − a a ( p − a · a ! . We maynow use the inequality a ! ≥ √ πa a +1 / exp( − a ) , Or indeed by non-elementary ones such as the classification result in [8, Theorem 1.2].
JOACHIM K ¨ONIG AND GICHEOL SHIN (a part of Stirling’s formula valid for all a ∈ N ), to obtain an upper bound √ π (cid:80) a | p − a )( p − a √ a ≤ (cid:80) a ≥ ( ep − ) a . Due to p ≥ log( n ) , the term for a = 1 gives a contribution O ( n ) ), whereasthe remaining sum is strictly smaller. This completes the proof of the claim of Step III.We may therefore assume from now on that σ and τ are such that, for any x ∈ S n , the group U above is either nonsolvable or imprimitive on O . To bound the proportion of pairs ( σ, τ ) forwhich U is solvable and imprimitive for at least one x ∈ S n , we return to the investigation of thecycle structure of σ .Let { O , . . . , O k } (with ∪ ki =1 O i = O ) be a block system of minimal block length preservedby the corresponding group U . Then U ≤ G (cid:111) H = ( G × · · · × G (cid:124) (cid:123)(cid:122) (cid:125) k times ) (cid:111) H , where G ≤ Sym ( O )is primitive (due to the minimality assumption), and H ≤ S k is transitive. Furthermore, recallthat some power of σ acts as a p -cycle on O , and this p -cycle necessarily lies in one component G (i.e., it permutes one block while fixing all the others pointwise). We assume U to be solvable,in which case G is solvable (and in particular affine) as well. But G also contains a p -cycle, so p < | O | − | O | ∈ { p + 1 , p + 2 } can be excluded as well, asalready shown in Step II.We have thus obtained | O | = p , so G ≤ AGL ( p ). We choose the ordering of the blocks O i such that O (cid:54) = { , . . . , p } Assume without loss of generality that σ permutes the m blocks O , . . . , O m cyclically ( m ≥ mp set O ∪ · · · ∪ O m by md , . . . , md r ( d i ∈ N ). Then σ m fixes the block O setwise, and induces apermutation of cycle lengths d , . . . , d r on this block. But this permutation needs to be containedin AGL ( p ), whence either r = 1 and d = p , or d = 1 and d = · · · = d r =: d . The first scenariois impossible since σ is assumed to contain only one p -cycle (1 , . . . , p ) and all other cycles of lengthcoprime to p . Setting a := p − d , we are left with the case that σ contains (among others) cyclesof the following lengths:i) One cycle of length p for some prime p ≥ log( n ) ,ii) One cycle of length m , and a cycles of length m ( p − a , for some a | p − m ∈ N . Step IV : We claim that the proportion of σ ∈ S n fulfilling i) and ii) above simultaneously isin O ( n ) ), which due to the above is enough to complete the proof.Indeed, due to Lemma 2.1, the proportion of σ ∈ S n fulfilling both i) and ii) is bounded fromabove by (cid:88) p ∈ P p ≥ log( n ) (cid:88) a | p − (cid:88) m ∈ N pm · ( m ( p − a ) a · a ! ≤ (cid:88) (cid:88) (cid:88) a a ( p − a +1 m a +1 · a ! == (cid:88) p ∈ P p ≥ log( n ) p − (cid:88) m ∈ N m + (cid:88) p ∈ P p ≥ log( n ) (cid:88) ≤ a | p − (cid:88) m ∈ N a a ( p − a +1 m a +1 a !The first sum is obviously, up to constant factor ≤ π , bounded from above by ∞ (cid:80) k =log( n ) k = O ( n ) ). To further bound the second sum, we once again use the inequality a ! ≥ √ πa a +1 / exp( − a ) to obtain an upper bound1 √ π (cid:88) p ≥ log( n ) (cid:88) ≤ a | p − (cid:88) m exp( a )( p − a +1 m √ a ≤ √ π (cid:88) m ≥ m (cid:88) p ≥ log( n ) exp(2)( p − (cid:88) a ≤ p √ a . NOTE ON INVARIABLE GENERATION OF NONSOLVABLE PERMUTATION GROUPS 7
The last sum over a is bounded from above by 2 √ p , so the double sum over p and a is, up toconstant factor, bounded from above by (cid:80) p ≥ log( n ) p − / = O ( n ) ) / ). Since the sum over m is absolutely bounded, the whole expression is in O ( n ) ), showing the claim and completingthe proof. (cid:3) Remark . If one only intents to show the “limit equals 1” part of Theorem 1.5, a few short-cuts in the above proof are possible. E.g., Step I may be shortened via replacing [15] by [7]which directly asserts that “most” permutations σ ∈ S n power to a cycle of prime length p ∈ [log n, (log n ) log log n ]; however, this comes at the cost of a worse asymptotic. Our ownasymptotic bound should also still be far from optimal (compare the experimental data in thenext section), mainly because the strong Condition (2) is far from necessary to invariably generatea nonsolvable subgroup.The assertion of Theorem 1.5 can easily be translated into a statement about the mean valueof the number of (independently chosen) random permutations required to invariably generate anonsolvable subgroup of S n . Corollary 3.2.
Let σ , σ , . . . be independent random permutations in S n or A n , and let N n :=min { r ∈ N | σ , . . . , σ r invariably generate a nonsolvable group } . Then lim n →∞ E ( N n ) = 2 .Proof. By Theorem 1.5, there exists (cid:15) n , tending to 0 with n → ∞ , such that P ( N n = 2) =1 − (cid:15) n . Furthermore, within the event N n >
2, the mean value is clearly bounded from above by E ( N n ) + 2 (simply by ignoring the first two permutations and seeking to invariably generate anonsolvable group with the remaining ones). Therefore we have E ( N n ) ≤ · (1 − (cid:15) n ) + (2 + E ( N n )) · (cid:15) n , or in other words E ( N n ) ≤ − (cid:15) n , which tends to 2 for n → ∞ . Conversely, N n cannot possiblybe smaller than 2, which shows the assertion. (cid:3) Some computational data and further directions
While Theorem 1.5 and Corollary 3.2 give assertions about the large degree limit, they saynothing about small values of n . We include some computational results on the number N n ofrandom permutations required to invariably generate a nonsolvable subgroup of S n , for somesmall n . Concretely, we calculated the probabilities P ( N n = 2). All computations were per-formed using Magma ([3]). The values are rounded to three decimal digits for convenience.These values actually seem to suggest a superlinear convergence of the “exceptional” proba-bility 1 − P ( N n = 2) (see Figure 1), although we emphasize that the sample size is too smallto draw definite conclusions. Note that 1 /n is a trivial lower bound for 1 − P ( N n = 2), sincethat is the probability to draw two n -cycles, which do not even invariably generate a non-cyclicsubgroup. A much deeper result by Blackburn, Britnell and Wildon ([2, Theorem 1.6]) statesthat the probability for two random permutations in S n to invariably generate a non-abelian sub-group is in fact still 1 − O (1 /n ). It would be interesting to find out whether this (or a similar)asymptotic bound might indeed still hold for the case of invariable generation of non-solvable Since the solvable subgroups of a group such as S are too numerous to be enumerated in full, the requiredcycle structures of maximally solvable subgroups were generated iteratively, using the obvious fact that such asubgroup must either be transitive or a direct product of two maximally solvable subgroups of smaller degree. JOACHIM K ¨ONIG AND GICHEOL SHIN
G P ( N n = 2) G P ( N n = 2) G P ( N n = 2) S S S S S S S S S S S S S S S S S S S S S Table 1.
Values for small symmetric groups
Figure 1.
Plot of points ( n, (1 − P ( N n = 2)) − )subgroups, although this should require a much more detailed analysis of cycle structures insolvable groups.Finally, here are a few thoughts on the case of groups G other than A n or S n . Obviously,a result such as Theorem 1.5 cannot be generalized to arbitrary families of nonsolvable groups G , and notably, the probability for two (or more) permutations of G to invariably generate anonsolvable subgroup of S n (in the sense of Definition 1.4) depends on the given permutationaction. E.g., if G acts in its regular permutation action, then every cycle type of G also occurs inthe cyclic group of order | G | , i.e., for polynomials with such a Galois group, one can never obtain anonsolvability certificate by relying solely on Dedekind’s criterion. But even for primitive groups G this can happen; e.g., G = A in its (degree 15) primitive action on 2-sets contains only cyclestructures which are also contained in the solvable group S (cid:111) AGL (5). For certain other groups, P ( N = 2) > P SL ( p ) ≤ S p +1 in its natural permutation action containscycle structures ( p,
1) and ( p +12 , p +12 ), which together invariably generate a doubly transitive and(for p ≥
5) nonsolvable subgroup of S p +1 . It would be interesting to know for which groupsDedekind’s criterion can detect nonsolvability (i.e., P ( N = r ) > r ), resp.,can detect it in two steps (i.e., P ( N = 2) > Appendix A. A strengthening
The proof of Theorem 1.5 can be adapted without many difficulties to show the followingstronger result.
NOTE ON INVARIABLE GENERATION OF NONSOLVABLE PERMUTATION GROUPS 9
Theorem A.1.
Let G = S n or G = A n . Then the proportion of random permutations σ, τ ∈ G which invariably generate a subgroup of S n with at least one (nonsolvable) alternating compositionfactor tends to as n → ∞ . To show the strengthening, we need the classification of primitive permutation groups con-taining a cycle (e.g., [8, Theorem 1.2]).
Proposition A.2.
Let G be a primitive group of degree m > not containing A m . Assumethat there exists an integer (cid:96) such that G contains an (cid:96) -cycle. Then one of the following holds: m = (cid:96) prime, and C (cid:96) ≤ G ≤ AGL ( (cid:96) ) , m = (cid:96) = ( q d − / ( q − for some d ≥ and prime power q ; and P GL d ( q ) ≤ G ≤ P Γ L d ( q ) , m = (cid:96) + 1 = q d for some d ≥ and prime power q ; and AGL d ( q ) ≤ G ≤ A Γ L d ( q ) , m = (cid:96) + 1 with (cid:96) prime, and P SL ( (cid:96) ) ≤ G ≤ P GL ( (cid:96) ) ; or m = (cid:96) + 2 = q + 1 for some prime power q ; and P GL ( q ) ≤ G ≤ P Γ L ( q ) . We now show how to adapt the proof of Theorem 1.5 to obtain Theorem A.1.
Proof of Theorem A.1.
Choose a prime p ( ≥ log( n ) ) as in Step I of the proof of Theorem 1.5.I.e., the permutation σ may be assumed to power to a p -cycle. Note next that in Cases 3) and5) of Proposition A.2 (with (cid:96) = p ), p is necessarily a Mersenne prime, which is excluded in StepII of the proof. In Case 2) of Proposition A.2, if d = 2, then p is necessarily a Fermat prime,which can be excluded in the same way; on the other hand, the sum of all reciprocals of numbers( q d − / ( q −
1) with d ≥ q a prime power, is convergent (e.g., bounded from above by thesome of all reciprocals of proper powers), hence for sufficiently large n , the sum of reciprocals ofall such numbers which are additionally ≥ log( n ) becomes arbitrarily small. We may thereforealter Step II of the proof of Theorem 1.5 to additionally demand that p is no such number.In particular, for the groups U in Step III and G in Step IV of the proof (which are primitivegroups containing a p -cycle by construction), Proposition A.2 then only leaves the possibility tobe alternating or symmetric of degree ≥ p , or to be contained in AGL ( p ) (already dealt with)or in P GL ( p ) in its primitive action on p + 1 points. But the cycle structures in the latter aresimilarly restricted as in AGL ( p ): they are all either ( p, d , . . . , d ), or ( d , . . . , d , , d of p + 1 and d of p −
1. Now using these cycle structures, Step III of the proofmay be carried out essentially as above, to show that with probability tending to 1, U mustbe alternating, symmetric or imprimitive. Finally, it remains to adapt Step IV to bound theprobability for σ to have both a p -cycle for some prime p ≥ log( n ) , and p +1 d cycles of the samelength md for some divisor d of p + 1 and some m ∈ N . This yields an upper bound essentiallyas in Step IV above, except that we lose one factor m . The reader may verify directly that theonly somewhat “critical” case is then d = p + 1 (i.e., σ has a p -cycle as well as one single cycleof length divisible by p + 1), which gives a contribution of at most (cid:88) p ∈ P p ≥ log( n ) p ( p + 1) n (cid:88) m =1 m ∈ O ( 1log( n ) · log( n )) = O (1 / log( n )) , which converges to 0. It therefore has been shown that with probability tending to 1, σ and τ invariably generate a subgroup with at least one alternating composition factor: namely eithercoming from a homomorphic image U acting primitively as the alternating or symmetric group This corresponds to the cycle type ( d , . . . , d ) of P GL ( p ) above; the estimate for the cycle types( d , . . . , d , ,
1) is analogous, and indeed easier. on some orbit; or from an imprimitive action of a homomorphic image U in which the blockkernel projects onto an alternating or symmetric group. (cid:3) Remark
A.3 . In the above proof, we have made no effort to retain the asymptotic of Theorem1.5, although this could be achieved; e.g., in the last step, since the only critical case is the onewhere there exists only a single cycle of length m ( p + 1), the imprimitive group U must have atotal number of m + 1 blocks of length p + 1 (one containing the p -cycle, and m further onespermuted cyclically by σ ). The image of U in the blocks action is then a transitive group ofdegree m + 1 containing an m -cycle, hence even 2-transitive and thus covered by PropositionA.2. Using this, the set of admissible values m can be restricted sufficiently. References [1] L. Bary-Soroker, G. Kozma,
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Department of Mathematics Education, Korea National University of Education, Cheongju, SouthKorea
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