aa r X i v : . [ m a t h . G R ] F e b Fixity of elusive groups and the polycirculant conjecture
Majid ArezoomandUniversity of Larestan, Larestan, 74317-16137, Iran
E-mail:[email protected]
Abstract.
Let G ≤ Sym(Ω) be transitive. Then G is called elusive onΩ if it has no fixed point free element of prime order. The 2 -closure of G , denoted by G (2) , Ω , is the largest subgroup of Sym(Ω) whose orbits onΩ × Ω are the same orbits of G . G is called 2-closed on Ω if G = G (2) , Ω .The polycirculant conjecture states that there is no 2-closed elusive group.In this paper, we study the fixity of elusive groups, where the fixity of G is the maximal number of fixed points of a non-trivial element of G .In particular, we prove that there is no 2-closed elusive solvable group offixity at most 5, a partial answer to the polycirculant conjecture. Keywords:
Fixity, elusive group, polycirculant conjecture.
Introduction
The concept of fixity was introduced by Rosen in 1980 to study of permu-tation groups of prime power order. Let G be a permutation group on a set Ω.We say that G has fixity f = f ( G ) if no non-trivial element of G fixes morethan f letters, and there is a non-trivial element of G fixing exactly f letter[11].Let G be a transitive permutation group. Then f ( G ) = 0 if and only if G is regular. Furthermore, f ( G ) = 1 if and only if G is a Frobenius group. Forsome applications of the fixity of permutation groups, we refer the reader to[8, 9, 11].A permutation g of a set Ω is said to be a derangement if it has no fixed-point on Ω, equivalently for each α ∈ Ω, α g = α . A transitive permutationgroup G ≤ Sym(Ω) is called elusive on Ω if it has no fixed point free elementof prime order. Let G be a permutation group on a set Ω. The 2 -closure of G , denoted by G (2) , Ω , is the largest subgroup of Sym(Ω) whose orbits on Ω × Ωare the same orbits of G . Then G is called 2-closed on Ω if G = G (2) , Ω .In 1981, Maruˇsiˇc asked whether there exists a vertex-transitive digraph with-out a non-identity automorphism having all of its orbits of the same length [10,Problem 2.4]. In 1988, independently, the above problem was again proposedby Jordan [6]. In the 15th British combinatorial conference, in 1995, Klin pro-posed a more general question in the context of 2-closed groups [7]: “Is there a2-closed transitive permutation group containing no fixed-point-free element ofprime order?” After decades, not only has a positive answer to Klin’s questionnot been found, based on the evidence, mathematicians conjecture that there isno positive answer to this question. This conjecture is known as polycirculantconjecture . Equivalently, the polycirculant conjecture states that no 2-closedtransitive permutation group is elusive. The conjecture is still open. We referthe reader to [1] to a survey on the recent results and future directions of thepolycirculant conjecture.In this paper, we study the fixity of elusive groups. Then we prove that everytransitive 2-closed solvable permutation group of fixity at most 5 confirms thepolycirculant conjecture. 2. Main Results
First we collect some notations we need later. Let G be a finite group and Ωbe a non-empty set. We denote the center of G and the set of all prime divisorsthe order of G by Z ( G ) and π ( G ), respectively. Also Sym(Ω) denotes the groupof all permutations on Ω. Let G acts on Ω and α ∈ Ω. We denote the stabilizerof α in G and the orbit of G containing α by G α and α G , respectively. AlsoFix Ω ( G ) denotes the fixed points of G on Ω, the set of all elements of Ω whichfixes by all elements of G . Furthermore, G Ω denotes the homomorphic imageof the action of G on Ω which is a subgroup of Sym(Ω). Recall that G is calleda permutation group on Ω if G is isomorphic to G Ω . For the notations andterminology not defined here, we refer the reader to [2].Let G be a permutation group on a set Ω. Recall that fixity of G is themaximal number of fixed points of a non-trivial element of G on Ω. We startwith the following lemma: Lemma 2.1.
Let G be a finite permutation group on Ω of fixity f ≥ , p > f a prime and α ∈ Ω . If p ∈ π ( G α ) then G α contains a Sylow p -subgroup of G .In particular, if G is transitive and contains a non-trivial normal p -subgroupthen p / ∈ π ( G α ) .Proof. Let p be a prime divisor of order of G α . Then there exists a Sylow p -subgroup P of G such that P α = 1. We claim that Z ( P ) ≤ G α . Supposetowards a contradiction, that our claim is not true. Then there exists g ∈ Z ( P )such that g / ∈ G α . Hence α, α g , . . . , α g p − are distinct elements of Fix Ω ( P α ), ixity of elusive groups and the polycirculant conjecture 3 because if α g i = α g j , for some 0 ≤ i < j ≤ p −
1, then g j − i ∈ G α and so( j − i, p ) = 1 implies that g ∈ G α , a contradiction. So P α = 1 implies that p ≤ f , a contradiction. This proves that Z ( P ) ≤ G α .Now we prove that P ≤ G α . Suppose, by a contrary, that P (cid:2) G α . Then x / ∈ G α for some x ∈ P . Hence, by a similar argument to the above paragraph, α, α x , . . . , α x p − are distinct. Now Z ( P ) ≤ G α implies that these elements areall in Fix Ω ( Z ( P )). Let y be a non-trivial element of Z ( P ). Then p ≤ | Fix Ω ( Z ( P )) | = | \ x ∈ Z ( P ) Fix Ω ( x ) | ≤ | Fix Ω ( y ) | ≤ f, which is a contradiction.Finally, suppose that G is transitive and N is a non-trivial normal p -subgroupof G . Then is contained in every Sylow p -subgroup of G . Now if p is a divisor of | G α | , then by the above paragraph, N ≤ G α . Since G is transitive, this impliesthat N is contained in any point-stabilizer of G which implies that N = 1,because G is a permutation group. This completes the proof. (cid:3) Corollary 2.2.
Let G be an elusive group on Ω with fixity f . If G contains anon-trivial normal p -subgroup then p ≤ f .Proof. Suppose, towards a contradiction, that p > f . Then by Lemma 2.1, p / ∈ π ( G α ), where α ∈ Ω. On the other hand, by [4, Lemma 2.1], π ( G ) = π ( G α )which is a contradiction. (cid:3) Corollary 2.3.
Let G be an elusive group on Ω of fixity f . Then f ≥ .Proof. Suppose, towards a contradiction, that f ≤
2. Since f = 0 if and onlyif G is regular on Ω, and since regular groups are not elusive, we have f = 1 , f = 1 if and only if G is a Frobenius group with Frobeniuscomplement G α on Ω, where α ∈ Ω. Since the Frobenius kernel of any finiteFrobenius group is a regular subgroup, we conclude that f = 2.Let α ∈ Ω. Then, by [4, Lemma 2.1], π ( G ) = π ( G α ). Hence, by Lemma 2.1, | Ω | = 2 k , for some k ≥
1, which contradicts [1, Lemma 2.6]. (cid:3)
Lemma 2.4.
Let G be an elusive group of fixity f ≥ on a finite set Ω . Let | Ω | = p n . . . p n k k , where n i ≥ and k ≥ . Then ( i ) for each ≤ i ≤ k , p i ≤ f , ( ii ) if p ∈ { p , . . . , p k } then every p -element in G is either fixed-point freeor fixes mp points, where ≤ m ≤ f /p .Proof. ( i ) Let α ∈ Ω and p be a prime divisor of | Ω | = | G : G α | . By Lemma2.1, if p > f then G α contains a Sylow p -subgroup of G . Hence p can not divide | Ω | , a contradiction.( ii ) Let p ∈ { p , . . . , p k } and x ∈ G be a p -element. Then x fixes at most f points. Since the length of any orbit of h x i on Ω is a power of p , by ( i ), | Ω | = | Fix Ω ( h x i ) | + lp , for some positive integer l . Hence p divides | Fix Ω ( h x i ) | . Majid Arezoomand
On the other hand, Fix Ω ( h x i ) = Fix Ω ( x ) which implies that x is either fixed-point free or fixes mp points, where 1 ≤ m ≤ f /p . (cid:3) Corollary 2.5.
Let G be an elusive group on Ω with fixity f . If | Ω | is odd,then f ≥ . In particular, if | G | is odd then f ≥ .Proof. Since | Ω | is a divisor of | G | , the second part is a direct consequenceof the first part. Suppose, by contrary, that f ≤
4. Then by Corollary 2.3, f ∈ { , } . Now by Lemma 2.4, | Ω | = 2 m n , where m, n ≥ m, n = 0. Let x be an element of order 3of G . Then, by Lemma 2.4, x is fixed-point free, which is a contradiction. (cid:3) For a finite group 1 = G , we denote the smallest prime divisor of | G | by p G .Then we have the following lemma. Lemma 2.6.
Let G be an elusive group on a finite set Ω with fixity f . Then | Ω | ≤ f min { | H | − p H − | = H E G } . Proof.
Let H be a non-trivial normal subgroup of G . Let Ω , . . . , Ω m be all H -orbits on Ω. By [2, Theorem 1.6. A], for i = j , H Ω i is permutation isomorphicto H Ω j . If H acts regularly on one orbit, then it acts regularly on all of itsorbits and so every element of prime order in H must be fixed-point free whichis a contradiction. Hence H does not act regularly on its orbits. Thus by [11,Lemma 2.6], | Ω | ≤ f ( | H | − / ( p H − (cid:3) Lemma 2.7.
Let G be an elusive group on a finite set Ω with fixity f . If G hasa non-trivial normal abelian subgroup N and p is the smallest prime divisor of | N | , then (1) | N | ≤ pf , (2) | Ω | ≤ f ( pf − p − ≤ f (2 f − .Proof. It is obvious that the fixity is at most f . On the other hand, by theproof of Lemma 2.6, N has no regular orbit on Ω. Hence, by [11, Lemma 2.7], | N | ≤ pf . The second part follows from Lemma 2.6. (cid:3) Corollary 2.8.
Let G be a -closed elusive solvable group of fixity f . Then f ≥ .Proof. Let G be a 2-closed elusive group on Ω. Suppose that N is a minimalnormal subgroup of G . Then N ∼ = Z kp for some k ≥
1, where p is a prime.Suppose, towards a contradiction, that f ≤
5. Then, by Lemma 2.7, | Ω | ≤ (cid:3) Corollary 2.9.
Let G be an elusive group of fixity f on Ω and N = 1 be anabelian normal subgroup of G of order p n p n . . . p n k k , where p < p < . . . < p k are primes, n i ≥ and k ≥ . Then (1) If k = 1 then n = 1 . Also ( n , . . . , n k ) = (1 , . . . , . ixity of elusive groups and the polycirculant conjecture 5 (2) p n + ... + n k − ≤ f . In particular, p k − ≤ f . (3) If f = 3 , then p = 2 or and N ∼ = Z or Z , respectively. (4) If f = 4 then p = 2 or . In the first case N is isomorphic to one ofthe groups Z , Z × Z p or Z × Z p , where p is an odd prime. In thelater case, N ∼ = Z .Proof. (1) If | N | = p or p p . . . p k then there exists a non-trivial normal cyclicsubgroup of G which contradicts [1, Lemma 2.20].(2) It is an immediate consequence of Lemma 2.7.(3) Since G is elusive, N is not cyclic by [1, Lemma 2.20]. Now (3) is aconsequence of (1) and (2).(4) Let f = 4. Then (2) implies that p = 2 or p = 3. If k = 1 then byLemma 2.7, N ∼ = Z . If k ≥ (cid:3) Corollary 2.10.
Let G be a transitive -closed permutation group on a set Ω of fixity and N = 1 be a normal p -subgroup of G . Then G admits fixed-pointfree element.Proof. Suppose, towards a contradiction, that G is elusive. Then, by Corollary2.9, we have N ∼ = Z or N ∼ = Z . Let α N be an orbit of N on Ω. Then | α N | ∈{ , p, p } , where p ∈ { , } . If | α N | = 1, then N = N α ≤ G α which impliesthat N = 1, a contradiction. If | α N | = p then N α = 1 which contradicts [1,Lemma 2.7]. Hence | α N | = p which contradicts [1, Theorem 2.11]. (cid:3) Acknowledgments
The author gratefully appreciate an anonymous referee for constructive com-ments and recommendations which definitely helped to improve the readabilityand quality of the article.
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