aa r X i v : . [ m a t h . G R ] S e p Finite groups with an automorphism of large order
Alexander Bors ∗ October 9, 2018
Abstract
Let G be a finite group, and assume that G has an automorphism of order atleast ρ | G | , with ρ ∈ (0 , ρ > /
2, then G is abelian, and if ρ > /
10, then G is solvable, whereas in general, the assumptionimplies [ G : Rad( G )] ≤ ρ − . , where Rad( G ) denotes the solvable radical of G .Furthermore, we generalize an example of Horoˇsevski˘ı to show that in finite groups,the quotient of the maximum automorphism order by the maximum automorphismcycle length may be arbitrarily large. The purpose of this paper is to study finite groups that may be viewed as “extreme”with respect to their maximum automorphism order. More generally, many authorshave studied finite groups satisfying “extreme” quantitative conditions of variouskinds. We mention the following examples: A variety of papers deals with finitegroups in which some automorphism raises some minimum fraction of elements tothe e -th power for e = − , ,
3, see [15, 16, 11, 12, 13, 14, 17, 4, 19, 7]. Wall classifiedthe finite groups G having more than | G | − G with more than | G | − ρ | G | , for some fixed ρ ∈ (0 , ρ > , then G isabelian [1, Theorem 1.1.7], and if ρ > , then G is solvable [2, Corollary 1.1.2(1)]. ∗ University of Salzburg, Mathematics Department, Hellbrunner Straße 34, 5020 Salzburg, Austria.E-mail: [email protected] author is supported by the Austrian Science Fund (FWF): Project F5504-N26, which is a part of theSpecial Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.2010
Mathematics Subject Classification : 20B25, 20D25, 20D45.
Key words and phrases:
Finite groups, Automorphisms, Abelian groups, Solvable groups, Solvable radical.
Furthermore, we showed that for any fixed value of ρ , the index of the solvable radicalRad( G ) in G is bounded from above in terms of ρ if G has such a long automorphismcycle [2, Theorem 1.1.1(1)].In this paper, we strengthen these results, replacing automorphism cycle lengthsby automorphism orders: Theorem 1.1.1.
Let G be a finite group.(1) If G has an automorphism of order greater than | G | , then G is abelian.(2) If G has an automorphism of order greater than | G | , then G is solvable.(3) For any ρ ∈ (0 , , if G has an automorphism of order at least ρ | G | , then [ G : Rad( G )] ≤ ρ E , where E = (log (6) − − = − . . . . . A few comments relating this to the results on cycle lengths. Automorphisms α of finite groups having a cycle of length ord( α ) (following the terminology in [6], suchcycles will be referred to as regular ) have been extensively studied by Horoˇsevski˘ıin [9]. He gave examples of automorphisms of finite groups without a regular cycle(i.e., whose order is larger than the largest cycle length). In Section 3, we willgeneralize one of Horoˇsevski˘ı’s examples to show that in finite groups, the quotientof the maximum automorphism order by the maximum automorphism cycle lengthmay be arbitrarily large.On the other hand, Horoˇsevski˘ı also gave conditions on finite groups G assuringthat every automorphism of G has a regular cycle, namely if G is either semisimple(i.e., has no nontrivial solvable normal subgroups) [9, Theorem 1] or nilpotent [9,Corollary 1]. Note that in view of the latter, Theorem 1.1.1 implies the following: Every automorphism α of a finite group G such that l := ord( α ) > | G | has a cycleof length l . In particular, for ρ > , the conditions “ G has an automorphism with acycle of length ρ | G | .” and “ G has an automorphism of order ρ | G | .” are equivalent,and by [1, Corollary 1.1.8], we obtain a complete classification of the pairs ( G, α )where G is a finite group and α an automorphism of G such that ord( α ) > | G | .We note that both in [1] and in [2], we did not only study automorphisms, but alarger class of permutations on finite groups, so-called bijective affine maps: Definition 1.1.2.
Let G be a finite group.(1) For an element x ∈ G and an endomorphism ϕ of G , the (left-)affine mapof G with respect to x and ϕ is the map A x,ϕ : G → G, g xϕ ( g ) .(2) The group of bijective affine maps of G (which are just those A x,ϕ where ϕ isan automorphism of G ) is denoted by Aff( G ) . We also had results for such maps, namely that a finite group G having a bijectiveaffine map with a cycle of length greater than | G | is solvable [2, Theorem 1.1.1(2)]and that [ G : Rad( G )] is bounded from above in terms of ρ in a finite group G havinga bijective affine map cycle of length at least ρ | G | . This can be strengthened in thesame way as the results on automorphisms: Theorem 1.1.3. (1) Let G be a finite group such that some A ∈ Aff( G ) has ordergreater than | G | . Then G is solvable.(2) Let ρ ∈ (0 , and let G be a finite group such that some A ∈ Aff( G ) has orderat least ρ | G | . Then [ G : Rad( G )] ≤ ρ E , where E = (log (30) − − = − . . . . . Finally, we remark that, just as for the results on cycle lengths, the constants , and in Theorems 1.1.1(1,2) and 1.1.3(1) respectively cannot be lowered further, asfollows by considering maximum automorphism orders in finite dihedral groups (forTheorem 1.1.1(1)) and in the alternating group A . By N + , we denote the set of positive integers. For a function f and a set M , wedenote by f [ M ] the element-wise image of M under f , and by f | M the restriction of f to M . The order of a group element g is denoted by ord( g ). We write N char G for“ N is a characteristic subgroup of G ”. The finite field with q elements is denoted by F q , and the logarithm with respect to a base c > c . We also use the followingnotation, most of which was already introduced in [1] and [2]: Notation. (1) Let X be a finite set, ψ a permutation on X . We denote by Λ( ψ ) thelargest cycle length of ψ and set λ ( ψ ) := | X | Λ( ψ ) .(2) For a finite group G , we define Λ( G ) := max α ∈ Aut( G ) Λ( α ) , λ ( G ) := | G | Λ( G ) , Λ aff ( G ) := max A ∈ Aff( G ) Λ( A ) and λ aff ( G ) := | G | Λ aff ( G ) .(3) For a finite group G , we denote by meo( G ) the maximum element order of G and set mao( G ) := meo(Aut( G )) and maffo( G ) := meo(Aff( G )) . This section gives a quick overview on some basic concepts and facts which we willneed.
Definition 2.1. (1) A finite dynamical system ( FDS ) is a finite set S togetherwith a function f : S → S , a so-called self-transformation of S . It is called periodic if and only if f is bijective.(2) For FDSs ( S, f ) and ( T, g ) , an (FDS) homomorphism between ( S, f ) and ( T, g ) is a function η : S → T such that η ◦ f = g ◦ η . The image of η ,denoted by im( η ) , is the FDS ( η [ S ] , g | η [ S ] ) An (FDS) isomorphism is a bijectiveFDS homomorphism.(3) If ( S , f ) , . . . , ( S r , f r ) are FDSs, their (FDS) product is defined as the FDS ( S ×· · ·× S r , f ×· · ·× f r ) , where f ×· · ·× f r maps ( s , . . . , s r ) ( f ( s ) , . . . , f r ( s r )) . [8, Sections 1–3] provides an introduction to the theory of FDSs. We will onlyneed the following proposition summarizing some elementary facts on cycle lengthsin FDS products: Proposition 2.2.
Let ( S , f ) , . . . , ( S r , f r ) be periodic FDSs.(1) The cycle length of ( s , . . . , s r ) ∈ S × · · · × S r under f × · · · × f r is the leastcommon multiple of the cycle lengths of the s i under f i .(2) If each f i has a regular cycle, then so does f × · · · × f r . Just as in [1], we will also work with the following notion:
Definition 2.3. A finite dynamical group ( FDG ) is a finite group G togetherwith an endomorphism ϕ of G . Hence any FDG is in particular an FDS, and an
FDG homomorphism is a mapbetween the underlying groups of two FDGs which is both a group homomorphismand an FDS homomorphism. We will need the following elementary result in theproof of Theorem 1.1.1(1):
Corollary 2.4.
Let ( G, α ) be a periodic FDG such that λ ( α ) > , and let ( Q, β ) bea homomorphic image of it. If ord( β ) = 1 , then the group Q is trivial.Proof. Note that ord( β ) = 1 implies that g − g ∈ ker η whenever g , g ∈ G lie onthe same cycle of α . Since α has a cycle of length greater than | G | by assumption,it follows that | ker η | > | G | , whence ker η = G by Lagrange’s theorem, and we aredone. mao( G ) / Λ( G ) The sole purpose of this section is to prove the following:
Proposition 3.1. sup G mao( G ) / Λ( G ) = ∞ , where G ranges over finite groups.Proof. Fix C ∈ N + . Denote by p , . . . , p C the first 3 C odd primes in increasingorder. For i = 1 , . . . , c , set B i := Z /p i Z , and let B := Q Ci =1 B i . Observe thatthose automorphisms of B that act by inversion on precisely one of the B i andidentically on the others generate an elementary abelian 2-subgroup E ≤ Aut( B ) of F -dimension 3 C . Let ψ : F C → E denote the F -isomorphism mapping a vector v to the automorphism ψ ( v ) =: α v of B acting identically on B i if the i -th componentof v is 0, and otherwise by inversion.Consider the 2-dimensional subspace U of F spanned by the vectors (0 , , t and (1 , , t together with the inclusion map ι : U ֒ → F . Form an external directsum U C of C copies of U . The product of C copies of ι (in the sense of Definition2.1(3)) is an embedding ι ′ : U C ֒ → ⊕ Ci =1 F . Furthermore, there is an isomorphism σ : ⊕ Ci =1 F → F C mapping the i -th standard basis vector, i = 1 , ,
3, of the j -thsummand, j = 1 , . . . , C , of the source to the (3( j −
1) + i )-th standard basis vectorof F C .Let W C denote the image of U C under the embedding σ ◦ ι ′ into F C , and let V C ⊆ E denote the image of W C under ψ . For i = 1 , . . . , C , denote by π i : F C → F theprojection onto the i -th component. Observe that W C (resp. V C ) has the followingtwo properties:(i) For each i = 1 , . . . , C , there exists v ∈ W C such that π i ( v ) = 1. Hence foreach i = 1 , . . . , C , there exists α v ∈ V C acting by inversion on B i .(ii) For each v ∈ W C , π i ( v ) = 0 for at least C values of i ∈ { , . . . , C } . Thuseach α v ∈ V C acts identically on at least C of the B i .Let G C be the subgroup of Hol( B ) = B ⋊ Aut( B ) generated by B and V C ; then G C = B ⋊ V C . We will be done once we have showed that mao( G C ) / Λ( G C ) ≥ C − . Let ξ be an automorphism of G C . Since the only elements of order p i in G C arethe nontrivial elements of B i , each B i (and hence B ) is ξ -invariant; fixing a nontrivialelement b i ∈ B i , we can write ξ ( b i ) = b k i i with k i ∈ ( Z /p i Z ) ∗ . Furthermore, sinceelements from different cosets of B in G C act identically on different collections of the B i , ξ restricts to a permutation on each coset of B . Hence for studying the dynamicsof ξ , we can partition G C into the cosets of B and study the dynamics on each coset.Let α v ∈ V C . Observe that if b is any element of B having nontrivial B i -component, where i is such that π i ( v ) = 0, then bα v does not have order 2. Hencewe can write ξ ( α v ) = Q Ci =1 b l i i α v with l i ∈ Z /p i Z and l i = 0 if π i ( v ) = 0. It is notdifficult to verify that the map Bα v → B, bα v b , is an isomorphism between theFDSs ( Bα v , ξ | Bα v ) and the FDS given by B = Q Ci =1 Z /p i Z together with the productof the affine self-transformations A l i ,k i of the Z /p i Z given by x k i x + l i . EachA l i ,k i has a regular cycle (so that ξ | Bα v has a regular cycle by Proposition 2.2(2)),and the order of A l i ,k i equals the order of k i ∈ ( Z /p i Z ) ∗ (which is a divisor of theeven number p i −
1) if k i = 1, and otherwise, it equals the order of l i ∈ Z /p i Z (whichis a divisor of p i ). Hence we always have ord(A l i ,k i ) ≤ p i , and for those i where π i ( v ) = 0, the order of A l i ,k i is a divisor of p i −
1. Since there are at least C such i by property (ii) above, this implies that the order (or largest cycle length) of ξ | Bα v is bounded from above by C − Q i,π i ( v )=0 ( p i − Q i,π i ( v )=1 p i ≤ C − Q Ci =1 p i .On the other hand, considering the inner automorphism ξ of G C with respect tothe element b · · · b C , ξ fixes each b i (so that k i = 1 for all i in the above notation),and ξ ( α v ) = Q i,π i ( v )=1 b i α v for v ∈ W C . In view of the above observations, thisimplies that every cycle length of ξ is a product of some of the p i , and by property(i) above, each p i occurs as a divisor of some cycle length. Hence ord( ξ ) = Q Ci =1 p i .It follows that mao( G C ) / Λ( G C ) ≥ Q Ci =1 p i / ( C − Q Ci =1 p i ) = 2 C − . Remark . As mentioned earlier, the groups G C described in the proof of Proposi-tion 3.1 are generalizations of an example given by Horoˇsevski˘ı, see [9, remarks afterCorollary 1]; Horoˇsevski˘ı’s example is our group G . mao rel and maffo rel In this section, we study the functions assigning to each finite group the quotient ofits maximum automorphism (resp. bijective affine map) order by the group order.We start with a very simple general lemma:
Lemma 4.1.
Let f be a function from the class G fin of finite groups to the interval (0 , ∞ ) such that f ( G ) = f ( G ) whenever G ∼ = G and f ( G/ Rad( G )) ≥ f ( G ) for allfinite groups G . Furthermore, assume that for finite semisimple groups H , f ( H ) → as | H | → ∞ ; more explicitly, fix a function g : (0 , ∞ ) → (0 , ∞ ) such that for any ρ ∈ (0 , ∞ ) , f ( H ) < ρ whenever H is a finite semisimple group with | H | > g ( ρ ) .Then for any ρ ∈ (0 , ∞ ) , if G is a finite group such that f ( G ) ≥ ρ , then [ G :Rad( G )] ≤ g ( ρ ) .Proof. By assumption, we have f ( G/ Rad( G )) ≥ f ( G ) ≥ ρ . Since G/ Rad( G ) issemisimple, this implies [ G : Rad( G )] = | G/ Rad( G ) | ≤ g ( ρ ) by choice of g . Lemma 4.1 summarizes our original approach to prove the weaker versions of The-orems 1.1.1(3) and 1.1.3(2) with cycle lengths instead of orders. Indeed, on the onehand, we observed that it follows from [1, Lemma 2.1.4] that λ (aff) ( G/N ) ≥ λ (aff) ( G )for any finite group G and N char G (implying the first assumption f ( G/ Rad( G )) ≥ f ( G ) for these two f ). On the other hand, assume that for some function f : G fin → (0 , ∞ ), we have | H | · f ( H ) ≤ | H | e for some e ∈ (0 ,
1) and all finite semisimplegroups H . Then clearly, if ρ ∈ (0 ,
1) and H is a finite semisimple group such that | H | > ρ / ( e − , then f ( H ) < ρ , whence g ( ρ ) from Lemma 4.1 can be chosen as ρ / ( e − . By [2, Lemma 3.4], we have Λ( H ) ≤ | H | log (6) and Λ aff ( H ) ≤ | H | log (30) for all finite semisimple groups H , thus explaining the exponents in [2, Theorem1.1.1].Moreover, we know by [2, Theorem 2.2.3] that mao( H ) = Λ( H ) and maffo( H ) =Λ aff ( H ) for all finite semisimple groups H . Hence by Lemma 4.1 and the remarksfrom the last paragraph, Theorems 1.1.1(3) and 1.1.3(2) are clear once we have provedthe following: Lemma 4.2.
Define functions mao rel , maffo rel : G fin → (0 , ∞ ) by mao rel ( G ) := | G | mao( G ) and maffo rel ( G ) := | G | maffo( G ) . Then mao rel ( G/ Rad( G )) ≥ mao rel ( G ) and maffo rel ( G/ Rad( G )) ≥ maffo rel ( G ) for all finite groups G . Before proving Lemma 4.2, we show:
Lemma 4.3.
Let B be a finite elementary abelian group, and fix β ∈ Aut( B ) . Then lcm x ∈ B ord(A x,β ) ≤ | B | .Proof. For x ∈ B , let sh β ( x ) := xβ ( x ) · · · β ord( β ) − ( x ) ∈ B . We observed in [2] thatord(A x,β ) = ord( β ) · ord(sh β ( x )). Hence the least common multiple in question iseither equal to ord( β ), which is bounded from above by | B | by [9, Theorem 2], or to p · ord( β ), where p is the prime base of | B | . Hence assume, for a contradiction, thatsome sh β ( x ) is nontrivial and that ord( β ) > p | B | . Considering the primary rationalcanonical form of β as an F p -automorphism (corresponding to a decomposition of B into a maximal number of subspaces that are cyclic for β ), we may assume byinduction that β can be represented by the companion matrix of P ( X ) k for someirreducible P ( X ) ∈ F p [ X ]. Note that all sh β ( x ) are fixed points of β , and so β hasa nontrivial fixed point by assumption. This implies that for some nonzero Q ( X ) ∈ F p [ X ] of degree less than deg( P ( X ) k ), we have X · Q ( X ) ≡ Q ( X ) (mod P ( X ) k ), orequivalently P ( X ) k | Q ( X ) · ( X − P ( X ) k ∤ Q ( X ), it follows that P ( X ) | X −
1, and thus P ( X ) = X − P ( X ) k (first proved by Elspas [5, Appendix II, 9], seealso [10, Theorem 3.11] and [8, Theorem 5 and remarks afterward]), it follows thatord( β ) = p ⌈ log p ( k ) ⌉ ≤ p k − = p | B | , a contradiction. Proof of Lemma 4.2.
We only prove that maffo rel ( G/ Rad( G )) ≥ maffo rel ( G ), as theargument for mao rel is similar. The proof is by induction on | Rad( G ) | . For theinduction step, fix A = A x,α ∈ Aff( G ) such that ord( A ) = maffo( G ). Following theargument in [9, proof of Theorem 2], we may fix a minimal α -invariant elementaryabelian normal subgroup B of G . By the induction hypothesis, it is sufficient to show that maffo rel ( G/B ) ≥ maffo rel ( G ). Denoting by ˜ A = A π ( x ) , ˜ α (where π : G → G/B is the canonical projection and ˜ α the induced automorphism on G/B ) the inducedaffine map of
G/B , we find that by [1, Lemma 2.1.4], every cycle length of A is aproduct of some cycle length of ˜ A with some cycle length of a bijective affine mapof B of the form A b,α | B . Hence the order of A divides the product of ord( ˜ A ) withlcm b ∈ B (A b,α | B ). In particular, by Lemma 4.3, ord( A ) ≤ ord( ˜ A ) · | B | . It follows thatmaffo rel ( G ) = | G | ord( A ) ≤ | G/B | ord( ˜ A ) ≤ maffo rel ( G/B ). As explained in Section 4, Theorems 1.1.1(3) and 1.1.3(2) follow from Lemmata 4.1and 4.2 as well as the remarks between them, and deriving Theorem 1.1.1(2) (resp.1.1.3(1)) from Theorem 1.1.1(3) (resp. 1.1.3(2)) is like in [2, proof of Corollary 1.1.2,Section 3]. Hence it only remains to prove Theorem 1.1.1(1).Fix an automorphism α of G such that ord( α ) > | G | . We prove that G is abelianby induction on | G | . For the induction step, observe that G cannot be semisimple,since otherwise, by [9, Theorem 1], α would have a regular cycle and hence G wouldbe abelian by [1, Theorem 1.1.7], contradicting its semisimplicity.Like in the proof of Lemma 4.2, following the argument in [9, proof of Theorem 2],we fix a minimal α -invariant elementary abelian normal subgroup B of G . We mayof course assume that B is proper in G . Denote by ˜ α the induced automorphism of G/B , set m := ord( ˜ α ), n := ord( α | B ) and denote by C the set of fixed points of α m in B . Horoˇsevski˘ı proceeded to show that either C = { } or C = B (by minimality of B ) and to derive upper bounds for ord( α ) in both cases, which imply that ord( α ) ≤ m · | G/B | in any case and thus m ≥ ord( α ) / | G/B | = | B | · ord( α ) / | G | > | B | , whence G/B is abelian by the induction hypothesis.In particular, we have G ′ ≤ B and λ ( ˜ α ) > by [9, Corollary 1]. Considerthe homomorphism ϕ : G → Aut( B ) corresponding to the conjugation action of G on B . Since B is abelian, we have B ≤ ker( ϕ ), and so there is a homomorphism ϕ : G/B → Aut( B ) such that ϕ ◦ π B = ϕ , where π B : G → G/B is the canonicalprojection.Now the kernel of ϕ consists by definition of those π B ( g ) ∈ G/B such that gB ⊆ C G ( B ). Clearly, since B is α -invariant, so ist C G ( B ), and thus ker( ϕ ) is ˜ α -invariant.It follows that there exists an automorphism α on the image ϕ ( G/B ) ≤ Aut( B ) suchthat the following diagram commutes: G/B G/Bϕ ( G/B ) ϕ ( G/B )˜ αϕ ϕα In other words, the FDG ( ϕ ( G/B ) , α ) is the image of the FDG ( G/B, ˜ α ) underthe FDG homomorphism ϕ : ( G/B, ˜ α ) → ( ϕ ( G/B ) , α ). By this definition of α , it is clear that ord( α ) | ord( ˜ α ) = m .We give an alternative definition of α . The element ϕ ( gB ) ∈ ϕ ( G/B ), which isby definition the restriction of conjugation by g to B , is mapped by α to α ( ϕ ( gB )) = ϕ ( ˜ α ( gB )) = ϕ ( α ( g ) B ), which is the restriction of conjugation by α ( g ) to B . Butthis implies that α is the restriction of conjugation by α | B in Aut( B ) to its subgroup ϕ ( G/B ). In particular, ord( α ) | ord( α | B ) = n .We now distinguish two cases. First, assume that B is cyclic. Then Aut( B ) isabelian, and so by the second definition of α , it is clear that α = id ϕ ( G/B ) . ByCorollary 2.4, this implies that ϕ is the trivial homorphism G/B → Aut( B ), and bydefinition of ϕ , this just means that B ≤ ζG . In particular, we have G ′ ≤ ζG , whence G is nilpotent of class 2. By [9, Corollary 1], this implies that λ ( α ) = ord( α ) > | G | ,and so G is abelian by [1, Theorem 1.1.7].Now assume that B ∼ = ( Z /p Z ) n for some prime p and n ≥
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