Cycle lengths in finite groups and the size of the solvable radical

0.33 MB·English

by  Alexander Bors

Cycle lengths in finite groups and the size of the solvable radical
∗
Alexander Bors
June 15, 2015
5
1
0
Abstract
2
n We prove the following: For any ρ (0,1), if a finite group G has an automorphism with a
∈
u cycle of length at least ρ G, then the index of the solvable radical Rad(G) in G is bounded from
J above in terms of ρ, and·s|uc|h a condition is strong enough to imply solvability of G if and only

2 if ρ > 1 . Furthermore, considering, for exponents e (0,1), the condition that a finite group G
1 have an10automorphism with a cycle of length at least∈ Ge, such a condition is strong enough to
| |
imply Rad(G) for G if and only if e > 1. We also prove similar results for a larger
] | | →∞ | | →∞ 3
R class of bijective self-transformations of finite groups, so-called periodic affine maps.
G
. 1 Introduction
h
t
a
m 1.1 Motivation and main results
[ In the author’s preprint [1], we studied how having an automorphism with a “long” cycle restricts

2 the structure of finite groups. One of the main results was that a finite group G with an automor-
v phism one of whose cycles has length greater than 1 G is necessarily abelian. For proving these
2| |
2 (and other) results, it turned out to be fruitful to study not only largest possible cycle lengths of
7
automorphismsoffinite groupsG,but alsoofamoregeneraltypeofbijective self-transformations,
1
namely maps A :G G ofthe formg g α(g) for somefixed g G andautomorphismα of
7 g0,α → 7→ 0 0 ∈
0 G. We called these maps periodic (left-)affine maps of G.
. Although most of the techniques introduced in [1] work under weaker assumptions as well, one
1
important idea (that an automorphism cycle, in a finite group G with G 3, of length at least
0
| | ≥
5 1 G must intersect with its pointwise inverse) makes explicit use of the fraction 1, and it is not
2| | 2
1 clear how one could derive similar results under the assumption that G have an automorphism
: cycle of length at least, say, 1 G.
v 3| |
i We will tackle this problem here by studying consequences of conditions on finite groups G of
X
theform“GhasanautomorphismcycleoflengthatleastρG”(“firstkind”)and“Ghasaperiodic
r affine map cycle of length at least ρG” (“second kind”) f|or|some fixed ρ (0,1), and also of the
a
| | ∈
form “G has an automorphism cycle of length at least Ge” (“third kind”) and “G has a periodic
| |
affinemapcycleoflengthatleast Ge”(“fourthkind”)forsomefixede (0,1). Ourmainresults,
| | ∈
all of which rely on the classification of finite simple groups (CFSG), are as follows:
Theorem 1.1.1. Let ρ (0,1) be fixed, let G be a finite group, and denote by Rad(G) the solvable
∈
radical of G. Then:
(1) If G has an automorphism cycle of length at least ρG, then [G : Rad(G)] ρE1, where
| | ≤
E = 1.778151....
1
−
∗Universityof Salzburg, Mathematics Department,HellbrunnerStraße 34, 5020 Salzburg, Austria.
E-mail: [email protected]
TheauthorissupportedbytheAustrianScienceFund(FWF):ProjectF5504-N26,whichisapartoftheSpecialResearch
Program “Quasi-Monte Carlo Methods: Theory and Applications”.
2010 Mathematics Subject Classification: primary: 20B25, 20D25, 20D45, secondary: 20D05, 20E22, 20G40, 37P99.
Key words and phrases: finitegroups, cyclestructure, solvable radical, semisimple groups
1
Alexander Bors Cycle lengths and the solvable radical
(2) If G has a periodic affine map cycle of length at least ρG, then [G : Rad(G)] ρE2, where
| | ≤
E = 5.906890....
2
−
So finite groups satisfying a condition of one of the first two forms are “not too far from
being solvable”. An interesting question is for which values of ρ such a condition actually implies
solvability. Notethatby[1,Theorem1.1.7],forconditionsofthefirstform,thisisthecasewhenever
ρ > 1. However, since solvability is a weaker condition than abelianity, one may hope to be able
2
to do better, and actually, we will prove:
Corollary 1.1.2. Let G be a finite group.
(1) If G has an automorphism cycle of length greater than 1 G, then G is solvable. On the other
10| |
hand, the alternating group has an automorphism cycle of length 6= 1 A .
A5 10| 5|
(2) If G has a periodic affine map cycle of length greater than 1 G, then G is solvable. On the
4| |
other hand, has a periodic affine map cycle of length 15= 1 .
A5 4|A5|
As for the conditions of the third and fourth kind mentioned above, we cannot expect results
as strong as Theorem 1.1.1 (see the discussion after Lemma 2.1.1), but we have the following:
Theorem 1.1.3. (1) Let ǫ > 0 be fixed. Then for every ξ > 0, there exists a constant K(ǫ,ξ)
such that for all finite groups G having an automorphism cycle of length at least G 13+ǫ, we have
| |
[G : Rad(G)] max(K(ǫ,ξ), G1−32ǫ+ξ). In particular, under a condition of the third kind with
≤ | |
e:= 1 +ǫ, for all ξ >0, we have G 32ǫ−ξ =o( Rad(G)) for G .
3 | | | | | |→∞
(2) Let ǫ > 0 be fixed. Then for every ξ > 0, there exists a constant K (ǫ,ξ) such that for all
aff
finitegroups G having a periodic affine map cycle of length at least G 23+ǫ, we have [G:Rad(G)]
| | ≤
max(K (ǫ,ξ), G1−3ǫ+ξ). In particular, under a condition of the fourth kind with e:= 2 +ǫ, for
aff | | 3
all ξ >0, we have G3ǫ−ξ =o( Rad(G)) for G .
(3) There exists a|seq|uence (G|n)n∈N of|finite|gr|o→ups∞Gn such that Rad(Gn) = 1 for all n N,
G for n , and for all n N, G has an automorphism| cycle of|length greater∈than
n n
| | → ∞ → ∞ ∈
Gn 13 and a periodic affine map cycle of length greater than Gn 32.
| | | |
We remarkthatwe canand willgive explicitdefinitions forK(ǫ,ξ), K (ǫ,ξ) andthe sequence
aff
(Gn)n∈N, see the proof of Theorem 1.1.3 at the end of Section 3.
1.2 Outline
In Section 2, we present the technical tools needed for proving our main results, some of which
were already introduced in [1] and are therefore given without proof here. None of them make use
of the CFSG. It turns out that using (part of) these tools, the proof of all of the main results can
be reduced to the proof of one technical lemma, namely Lemma 3.1, which we will call the “main
lemma”. It is a statement about maximum cycle lengths of automorphisms and of periodic affine
mapsoffinitenonabeliancharacteristicallysimplegroups,anditsproofwillusetheCFSG.Section
3 shows how the main lemma implies all the main results, and Section 4 consists of the proof of
the main lemma.
1.3 Notation and terminology
Forthe readers’convenience,we explainthoseparts ofournotationthat maybe nonstandard. We
denote by N the set of natural numbers (von Neumann ordinals, including 0), and by N+ the set
of positive integers. The image of a set M under a function f is denoted by f[M]. The identity
function on a set M is denoted by id , and the symmetric group on M is denoted by , except
M M
S
whenM isanaturalnumbern,inwhichcaseweset := . Similarly,foranaturalnumber
n {1,...,n}
S S
n, is the alternating group on 1,...,n .
n
A { }
Let G be a group. For an element r G, we denote by τ : G G,g rgr−1 the inner
r
∈ → 7→
automorphism of G with respect to r. The centralizer and normalizer of a subset X G are
⊆
denoted by C (X) and N (X) respectively. As in Theorems 1.1.1 and 1.1.3, Rad(G) denotes the
G G
solvable radical of G. For linguistical simplicity, we will frequently use the following notation, see
also [1, Definitions 1.1.1, 2.1.1 and 2.1.2] as well as [5]:
2
Alexander Bors Cycle lengths and the solvable radical
Definition 1.3.1. (1) Let ψ be a permutation of a finite set X. We denote by Λ(ψ) the maximum
length of one of the disjoint cycles into which ψ decomposes, and set λ(ψ):= 1 Λ(ψ).
|X|
(2) For a finite group G, we set Λ(G):=max Λ(α) and λ(G):= 1 Λ(G).
α∈Aut(G) |G|
(3) For a finite group G, the group of periodic left-affine maps of G is denoted by Aff(G). We set
Λ (G):=max Λ(A) and λ (G):= 1 Λ (G).
aff A∈Aff(G) aff |G| aff
(4)For afinitegroupG, wedenotebymeo(G)themaximumelement order ofGandsetmao(G):=
meo(Aut(G)), the maximum automorphism order of G.
We also use some notation and terminology from the theory of finite dynamical systems:
Definition1.3.2. (1) Afinite dynamical system, abbreviated henceforth byFDS,is afiniteset
X together with a map f :X X (a so-called self-transformation of X). It is called periodic
→
if and only if f is bijective.
(2) If (X ,f ),...,(X ,f ) are FDSs, their product is defined as the FDS (X X ,f
1 1 r r 1 r 1
×···× ×
f ), where f f is the self-transformation of X X mapping (x ,...,x )
r 1 r 1 r 1 r
···× ×···× ×···× 7→
(f (x ),...,f (x )).
1 1 r r
(3) If (X,ψ)is a periodic FDS andx X, wedenote thelength of thecycle of x under ψ bycl (x).
ψ
∈
Finally, in this paper, exp mostly denotes the exponent of a group, although in the definition
of Ψ in Subsection 2.4, it denotes the naturalexponential function. log always denotes the natural
logarithm, and for c>1, the logarithm with base c is denoted by log .
c
2 Some tools
2.1 Lemmata concerning maximum cycle lengths
Lemma 2.1.1 below was used in the proof of [1, Lemma 2.1.6], of which Lemma 2.1.2 is a part.
Lemma 2.1.1. Let (X ,ψ ),...,(X ,ψ ) be periodic FDSs, and let x = (x ,...,x ) X
1 1 r r 1 r 1
∈ ×
X . Then cl (x) = lcm(cl (x ),...,cl (x )). In particular, Λ(ψ ψ )
···× r ψ1×···×ψr ψ1 1 ψr r 1 ×···× r ≤
Λ(ψ ) Λ(ψ ).
1 r
···
We remark that by Lemma 2.1.1, any condition on finite groups G of the form λ(G) f(G),
where f : N+ [0,1] is such that f(n) 0 for n , is not strong enough to imply≥that| th|e
→ → → ∞
index[G:Rad(G)]isboundedfromabove. Indeed,undersucha condition,anyfinite groupG (in
0
particular, any nonabelian finite simple group G ) may occur as a direct factor of G. To see this,
0
let p be a prime which is so large that f(pG ) 1 . Considering the product automorphism
| 0| ≤ 2|G0|
id α of G := G Z/pZ, where α Aut(Z/pZ) is the multiplication by any primitive root
G0× 0 × ∈
modulo p, it is not difficult to see by Lemma 2.1.1 that
p 1 1 1 1
λ(G) λ(id α)= − =(1 ) f(G).
≥ G0× pG − p · G ≥ 2G ≥ | |
0 0 0
| | | | | |
As in [1], we say that a family (G ) of groups has the splitting property if and only if for
i i∈I
every automorphism α of G , there exists a family (α ) such that α is an automorphism
i∈I i i i∈I i
of Gi for i∈I, and α((gi)iQ∈I)=(αi(gi))i∈I for all (gi)i∈I ∈ i∈IGi.
Lemma 2.1.2. Let (G ,...,G ) be a tuple of finite groups wQith the splitting property. Then:
1 r
(1) Λ(G G ) Λ(G ) Λ(G ).
1 r 1 r
×···× ≤ ···
(2) For every periodic affine map A of G G , there exists a tuple (A ,...,A ) such that
1 r 1 r
×···×
A Aff(G ) for i = 1,...,r and A = A A . In particular, Λ (G G )
i i 1 r aff 1 r
∈ × ··· × × ··· × ≤
Λ (G ) Λ (G ).
aff 1 aff r
···
The following is a part of [1, Lemma 2.1.4]:
Lemma 2.1.3. Let G be a finite group, N a characteristic subgroup of G. Then:
(1) Λ(G) Λ (N) Λ(G/N), or equivalently, λ(G) λ (N) λ(G/N). In particular, λ(G/N)
aff aff
≤ · ≤ · ≥
λ(G).
(2) Λ (G) Λ (N) Λ (G/N), or equivalently, λ (G) λ (N) λ (G/N). In particular,
aff aff aff aff aff aff
≤ · ≤ ·
λ (G) min(λ (N),λ (G/N)).
aff aff aff
≤
3
Alexander Bors Cycle lengths and the solvable radical
WewillnowprovesomemoreresultsthatareusefulforthestudyofΛ -valuesoffinitegroups.
aff
For a more concise formulation, we define:
Definition 2.1.4. Let G be a finite group, x G, α an automorphism of G, n N+.
(1) The element sh(n)(x):=xα(x) αn−1(x)∈ G is called the n-th shift of x∈under α.
α ··· ∈
(2) The element sh (x):=sh(ord(α)) G is called the shift of x under α.
α α ∈
The following calculation rules for shifts are easy to show:
Lemma 2.1.5. Let G be a finite group, x G, α an automorphism of G.
∈
(1) α(sh (x))=xsh (x)x−1.
α α
(2) If d∈N+ is such that clα(x)|d|ord(α), then shα(x)=shα(d)(x)orddα.
Definition2.1.4ismotivatedbythefollowing: Itiswell-knownthatthereisnaturalisomorphism
between Aff(G), the product, inside , of the image of the left regular representation of G with
G
Aut(G), and the holomorph of G, HoSl(G)=G⋊Aut(G). The isomorphism is simply given by the
mapAff(G) Hol(G),A (x,α). Itisthereforeclearthatord(α) ord(A )forallx Gand
x,α x,α
→ 7→ | ∈
all α Aut(G), and thus ord(A ) = ord(α) ord(Aord(α)). However, easy computations reveal
∈ x,α · x,α
that under said natural isomorphism, Aord(α) corresponds to the element sh (x) G. This shows
x,α α ∈
thatingeneral,wehavethefollowingformulaforcomputingordersofperiodicaffinemapsoffinite
groups:
ord(A )=ord(α) ord(sh (x)).
x,α α
·
When ψ is a permutation of a finite set X and n N+, we say that an orbit O of the action
of ψ on X induces an orbit O˜ of ψn (or that O˜ stem∈s from O) if and only if O˜ O, in which
case O˜ = 1 O. Every orbitof ψ induces an orbitof ψn, and every orbit of⊆ψn stems from
| | gcd(n,|O|)| |
precisely one orbit of ψ.
Lemma 2.1.6. Let G be a finite group, x G, α an automorphism of G. Then every cycle length
∈
of Ax,α is divisible by LG(x,α) := ord(shα(x))· ppνp(ord(α)), where p runs through the common
prime divisors of ord(shα(x)) and ord(α). In parQticular, LG(x,α) G.
|| |
Proof. Every orbit of Aord(α), the left multiplication by sh (x) in G, has size ord(sh (x)), so
x,α α α
certainly every cycle length of A is divisible by ord(sh (x)). In particular, if p is a common
x,α α
primedivisoroford(shα(x))andord(α),andO isanyorbitofAx,α,thenp O,butpνp(ord(shα(x)))
|| |
still divides O˜ , where O˜ is the orbit of Aord(α) induced by O. This is only possible if O actually
| | x,α | |
is divisible by pνp(ord(shα(x)))+νp(ord(α)), and the assertion follows.
Lemma 2.1.7. Let G be a finite group, x,r G. Then x−1r C (sh (x)). In particular, if, for
∈ ∈ G τr
some subgroup H G, C (sh (x)) H, then x H if and only if r H.
≤ G τr ⊆ ∈ ∈
Proof. This follows immediately from rsh (x)r−1 = τ (sh (x)) = xsh (x)x−1, where the first
τr r τr τr
equality is by the definition of τ and the second by Lemma 2.1.5(1).
r
Lemma 2.1.8. (1) Let G be a finite centerless group, r,s G. Set x := sr−1. Then sh (x) =
∈ τr
sord(r). In particular, ord(A )=lcm(ord(s),ord(r)).
x,τr
(2) Let G be any finite group, r,s ∈ G, x as in point (1). Then shτr(x) = sord(τr)·r−ord(τr). In
particular, if gcd(ord(r),ord(s))=1, then ord(A )=ord(s) ord(r).
x,τr ·
Proof. An easy induction on n N+ proves that in both cases, we have sh(n)(x)=snr−n. There-
∈ τr
fore, we have sh (x)=sord(r) under the assumptions of point (1). This implies that
τr
ord(s)
ord(A )=ord(τ ) ord(sh (x))=ord(r) =lcm(ord(s),ord(r)),
x,τr r · τr · gcd(ord(s),ord(r))
provingthe statementofpoint(1). The proofofpoint(2)issimilar, usingthatr−ord(τr) ζGand
∈
that the orderof a product oftwo commuting elements with coprime ordersis the productof their
orders.
4
Alexander Bors Cycle lengths and the solvable radical
2.2 Some results on finite semisimple groups
In this Subsection, for the readers’ convenience, we first briefly recall some basic facts on finite
semisimple groups (finite groups without nontrivial solvable normal subgroups) which we will
need later, following mostly the exposition in [9, pp. 89ff.]. Afterward, we generalize a result
of Horoˇsevski˘ı on largest cycle lengths of automorphisms of finite semisimple groups to periodic
affine maps of such groups.
Any groupG has a unique largestnormalcenterless CR-subgroup, the centerless CR-radicalof
G, which we denote by CRRad(G). From now on, assume that G is finite and semisimple. Then
CRRad(G) coincides with Soc(G), the socle of G. G canonically embeds into Aut(Soc(G)) by its
conjugation action (which shows that for any finite centerless CR-group R, there are only finitely
many isomorphism types of finite semisimple groups G such that Soc(G) =R), and the image G∗
∼
of this embedding clearly contains Inn(Soc(G)). Conversely, for every finite centerless CR-group
R, any group G such that Inn(R) G Aut(R) is semisimple.
If S ,...,S are pairwise nonis≤omor≤phic nonabelian finite simple groups, and n ,...,n N+,
1 r 1 r
then the tuple (Sn1,...,Snr) has the splitting property. In particular, Aut(Sn1 S∈nr) =
Aut(Sn1) A1ut(Snr).rThe structure ofthe automorphismgroupsoffinite n1on×ab·e·l·ia×nchrarac-
1 ×···× r
teristically simple groups (powers of finite nonabelian simple groups) can be described by permu-
tational wreath products. More precisely, Aut(Sn) =Aut(S) for any finite nonabelian simple
n
group S and any n N+. ≀S
∈
Rose [10, Lemma 1.1]observedthat, in generalizationof the embedding of G into Aut(Soc(G))
for finite semisimple groups G, if G is any group, and H a characteristic subgroup of G such that
C (H)= 1 , then G embeds into Aut(H) by its conjugation action on H, and, viewing G as a
G G
{ }
subgroup of Aut(H), Aut(G) is canonically isomorphic to N (G). This implies, among other
Aut(H)
things, that automorphism groups of finite centerless CR-groups are complete.
Letus nowturnto the aforementionedtheoremofHoroˇsevski˘ı. Followingthe terminologyfrom
[4], we define:
Definition 2.2.1. Let ψ be a permutation of a finite set. A cycle of ψ whose length equals ord(ψ)
is called a regular cycle of ψ.
Thus a permutation ψ of a finite set has a regular cycle if and only if Λ(ψ) = ord(ψ). In the
case of periodic affine maps A of finite groups G, the order is often easier to compute than the
Λ-value, since for computing the order, one can work with the compact representation A = A
x,α
for appropriate x G and α Aut(G), and composition of periodic affine maps translates, on
∈ ∈
the level of the compact representations, into some simple manipulations (by the isomorphism
Aff(G) Hol(G)mentionedabove),withouttheneedto“spreadout”theentireelementstructure
→
of G to determine the cycle lengths of the elements of G under A.
In view of this, it would be nice to know at least for some classes of finite groups G that all
periodic affine maps of G have a regular cycle to make computation of Λ- and Λ -values easier.
aff
Indeed, Horoˇsevski˘ıproved:
Theorem 2.2.2. ([6, Theorem 1]) Let G be a finite semisimple group. Then every automorphism
of G has a regular cycle.
We will extend this to:
Theorem 2.2.3. Let G be a finite semisimple group. Then every periodic affine map of G has a
regular cycle.
OurproofofTheorem2.2.3ismostlyanadaptationofHoroˇsevski˘ı’sproofofTheorem2.2.2,with
the arguments getting slightly more complicated because of the more general situation. However,
at one point, our proof significantly differs from the one of Horoˇsevski˘ı,using the recent result [4,
Theorem 3.2] to settle one important case. Just like Horoˇsevski˘ı,we use the following:
Lemma 2.2.4. Let X be a finite set, ψ , p a prime such that p2 ord(ψ). The following are
X
∈S |
equivalent:
(1) ψ has a regular cycle.
(2) ψp has a regular cycle.
5
Alexander Bors Cycle lengths and the solvable radical
Proof. See [6, Lemma 1]. The assumption there that ψ (called φ there) is an automorphism of a
finite group is not needed.
Beforewecontinuewiththenextlemma,aquickreminderandaneasyobservation: Recallthat
for a groupG, an automorphismα of G, and a normalsubgroupNEG, α-admissibility of N (i.e.,
the property that α(N) = N) is equivalent to the existence of an automorphism α˜ of G/N such
that, denoting by π : G G/N the canonical projection, π α = α˜ π. In this case, α˜ is unique
→ ◦ ◦
and is called the automorphism of G/N induced by α. More generally, if, for some permutation ψ
of G, there exists a permutation σ of G/N such that π ψ = σ π, we still call σ induced by ψ.
◦ ◦
It is not difficult to see that for any group G, any N EG and any periodic affine map A = A
x,α
of G, A induces a permutation A˜ of G/N if and only if N is α-admissible, and in this case, A˜ is a
periodic affine map of G/N; actually, A˜=A .
π(x),α˜
Lemma 2.2.5. Let G be a group, BEG, A a periodic affine map of G such that A =id . Then
|B B
C (B)EG, and A induces the identity map in G/C (B).
G G
Proof. Ingeneral,for allx Gandα Aut(G), it followsimmediately fromthe definitionofA
x,α
∈ ∈
that A (1 ) = x. Since A(1 ) = 1 by assumption, A thus actually is an automorphism of G,
x,α G G G
so the claim follows from [6, Lemma 2].
Lemma 2.2.6. Let X ,...,X be finite sets, ψ , i=1,...,n, a permutation of X with a regular
1 n i i
cycle. Then ψ ψ has a regular cycle.
1 n
×···×
One additional easy observation which we will need is the following:
Lemma 2.2.7. Let G be a group, A = A a periodic left affine map of G such that fix(A) = .
x,α
6 ∅
Then fix(A) is a left coset of the subgroup fix(α) G.
≤
Proof. For all g G, we have that g fix(A) if and only if xα(g)=g, or x=gα(g)−1. Therefore,
∈ ∈
if we fix f fix(A), then fix(A) can be desribed as g G gα(g)−1 = fα(f)−1 = g G
∈ { ∈ | } { ∈ |
g−1f fix(α) =ffix(α).
∈ }
Proof of Theorem 2.2.3. Theproofisbyinductionon G, theinductionbase G =1being trivial,
| | | |
with an inner induction on ord(A), the induction base ord(A)=1 being trivial. For the induction
step, assume that A = A is a periodic affine map of the finite semisimple group G. To show
x,α
that A has a regular cycle, we make a case distinction:
1. Case: Gissimple. Thiscaseisbycontradiction,soassumethatAdoesnothavearegularcy-
cle. NotethatbyLemma2.2.4andtheinductionhypothesis,ord(A)thenmustbesquarefree,
sayord(A)=p p ,withthep pairwisedistinctprimes. Sincebytheinductionhypothesis,
1 r i
···
Ap1 hasa cycle oflengthord(Ap1)=p2 pr, but Ahas noregularcycle, Amust alsohavea
···
cycleoflengthp p ,whichimpliesp p < G. NownotethatbytheassumptionthatA
2 r 2 r
··· ··· | |
does not have a regularcycle, we have G r fix(AQj6=ipj). By Lemma 2.2.7, denoting by
⊆ i=1
αi the underlyingautomorphismofAQj6=ipjS,wehave fix(AQj6=ipj) = fix(αi),andsothere
| | | |
mustexisti 1,...,r suchthat[G:fix(α )] r (otherwise, Gcouldnotbe coveredby the
i
∈{ } ≤
r fixed point sets above). But since G is simple, this implies that G r! p p < G,
2 r
| | ≤ ≤ ··· | |
a contradiction.
2. Case: G is characteristically simple, but not simple. Let S be a nonabelian finite simple
group and n 2 such that G = Sn. α is an element of the permutational wreath product
≥ ∼
Aut(S) , i.e., α is a composition (α α ) ψ, where each α is an automorphism
n 1 n i
≀S ×···× ◦
of S and ψ is a permutation of coordinates on Sn. Writing x = (x ,...,x ), and denoting
1 n
by µ the left multiplication by x in Sn, it follows that A = µ ((α α ) ψ) =
x x 1 n
◦ ×···× ◦
((µ µ ) (α α )) ψ = (A A ) ψ. This proves that
x1 ×···× xn ◦ 1 ×···× n ◦ x1,α1×···× xn,αn ◦
A Aff(S) (actually,wejustprovedthatAff(Sn)=Aff(S) ). Byinductionhypothesis,
n n
∈ ≀S ≀S
everypermutationfromAff(S)hasaregularcycle,andsoby[4,Theorem3.2],Ahasaregular
cycle.
6
Alexander Bors Cycle lengths and the solvable radical
3. Case: G is completely reducible, but not characteristicallysimple. Let S ,...,S be pairwise
1 r
nanondinsoomteotrhpahticrnon2abbyelaiasnsufimnpitteiosnim. pSliencgero(Supns1,,.n.1.,,.S..n,rn)rha∈sNth+essupclhitttihnagtpGro∼=peSrt1ny1,×by··L·×emSmrnra,
2.1.2(2), A can≥be written as a product of 1periodicraffine maps over the single Sni, each of
i
whichhasaregularcyclebytheinductionhypothesis,andsoAhasaregularcyclebyLemma
2.2.6.
4. Case: G is not completely reducible. Set B := Soc(G), and note that B is proper in G
and C (B) = 1 . Denote by A˜ the periodic affine map of G/B induced by A, and let k
G G
denote the leng{th o}f the identity element of G/B under A˜. Set A :=Ak. Then A restricts
0 0
to a periodic affine map of B, so by the induction hypothesis, A has a cycle of length
0|B
n := ord(A ); fix an element x B such that cl (x) = n. Now An acts identically in
0|B ∈ A0 0
B, and thus by Lemma 2.2.5 also in G = G/C (B). This means that n = ord(A ), and so
∼ G 0
ord(A) k n. But clearly, cl (x) = k n, since k divides the cycle length under A of any
A
≤ · ·
element from B. Therefore, ord(A)=k n and A has a regular cycle.
·
Corollary 2.2.8. (1) Let G be a finite semisimple group. Then:
(i) Λ(G)=mao(G).
(ii) Λ (G)=meo(Hol(G)).
aff
(2) Let R be a finite centerless CR-group. Then:
(i) Λ(Aut(R))=mao(R).
(ii) Λ (Aut(R))=meo(Hol(Aut(R))).
aff
Proof. For(1): (i)isanimmediateconsequenceofTheorem2.2.3,and(ii)alsofollowsfromTheorem
2.2.3 and the fact that Aff(G)=Hol(G).
∼
For (2): As for (i), note that Aut(R) is semisimple, and so by (1,i), we have Λ(Aut(R)) =
mao(Aut(R)) = meo(Aut(Aut(R))) = meo(Aut(R)) = mao(R), where the second-to-last equality
follows from the completeness of Aut(R). (ii) just is a special case of (1,ii).
2.3 Upper bounds on element orders in wreath products
We will need upper bounds on meo(G) and mao(G) for finite semisimple groups G. To this end,
someboundsonordersofelementsinwreathproductsingeneralcomeinhandy. Beforeformulating
and proving Lemma 2.3.2 below, we introduce the following notation and terminology:
Definition 2.3.1. Let G be a finite group, n N+, and ψ .
n
∈ ∈S
(1) Letg =(g ,...,g ) Gn. For i=1,...,n, wedefineel(ψ)(g):=g g g G.
1 n ∈ i i ψ−1(i)··· ψ−clψ(i)+1(i) ∈
Alternatively, one can describe el(ψ)(g) as the image of sh(clψ(i))(g) Gn G under the
i τψ ∈ ≤ ≀ Sn
projection π :Gn G onto the i-th component.
i
→
(2) We denote the set of orbits of the action of ψ on 1,...,n by Orb(ψ).
{ }
(3) An assignment to ψ in G is a function β : Orb(ψ) G. For such an assignment β, we
→
ord(β)
define its order to be the least common multiple of the numbers ord(β(O) |O| ), where O runs
through Orb(ψ).
Lemma2.3.2. LetGbeafinitegroup, n N+, denotebyπ :G thecanonical projection,
n n
∈ ≀S →S
and let ψ .
n
(1) Let g =∈S(g ,...,g ) Gn and consider the element x:=(g,ψ) Gn⋊ = G . Then for
1 n n n
∈ ∈ S ≀S
ord(ψ)
i=1,...,n, the i-th component of xord(ψ) Gn equals el(ψ)(g)clψ(i).
∈ i
(2) In particular, the maximum order of an element x G such that π(x) = ψ equals the
n
∈ ≀S
product of ord(ψ) with the maximum order of an assignment to ψ in G and is bounded from above
by ord(ψ) meo(G|Orb(ψ)|).
·
Proof. For (1): We may assume that G is nontrivial. Fix i, and denote by π : Gn G the
i
→
projection onto the i-th component. It is clear that xord(ψ) = sh (g) (where the shift is formed
τψ
7
Alexander Bors Cycle lengths and the solvable radical
inside G and τ is the inner automorphism of G with respect to ψ), whence π (xord(ψ))=
n ψ n i
≀S ≀S
π (sh (g)). But the i-th component of sh (g) only depends on the components of g whose
i τψ τψ
indices are from the orbit O of i under ψ, so if we denote by g˜ the element of Gn which has the
i
same entries as g in the components whose indices are in O but all other entries equal to 1 ,
i G
we have π (xord(ψ)) = π (sh (g˜)). Now note that cl (i) is a multiple of cl (g˜) and a divisor of
i i τψ ψ τψ
ord(ψ)=ord(τ ), which gives us, by an application of Lemma 2.1.5(2),
ψ
ord(ψ) ord(ψ)
πi(xord(ψ))=πi(shτψ(g˜))=πi(shτ(cψlψ(i))(g˜)clψ(i))=πi(shτ(cψlψ(i))(g˜))clψ(i) =
ord(ψ) ord(ψ)
πi(shτ(cψlψ(i))(g))clψ(i) =el(iψ)(g)clψ(i).
For (2): For any element x G of the form (g,ψ), we have ord(x) = ord(ψ) ord(xord(ψ)),
n
∈ ≀S ·
ord(ψ)
where,by (1),the secondfactoris the leastcommonmultiple ofthe numbersord(el(ψ)(g)clψ(i))for
i
i=1,...,n. Fix a set of representatives of the orbits of ψ, which is in canonical bijection with
R
Orb(ψ). It is not difficult to see that if i,j 1,...,n are from the same orbit under ψ, then
∈ { }
ord(ψ) ord(ψ)
el(ψ)(g)clψ(i) and el(ψ)(g)clψ(j) are conjugate in G and thus have the same order,so ord(xord(ψ)) is
i j
equal to just the least common multiple of the numbers ord(el(ψ)(g)ocrldψ((ψi))) for i . Therefore,
i ∈ R
composing the canonical bijection Orb(ψ) with the function G,i el(ψ)(g) gives an
→ R R → 7→ i
assignmenttoψ inGwhoseordercoincideswithord(xord(ψ)). Conversely,ifanyassignmentβ toψ
inGisgiven,bychoosingthe componentsg ,...,g ofGsuchthatforallO Orb(ψ) thereexists
1 n
∈
i O such that g g g = β(O), we can assure that ord((g,ψ)ord(ψ)) = ord(β).
∈ i ψ−1(i)··· ψ−clψ(i)+1(i)
This proves the claim.
2.4 Landau’s and Chebyshev’s function
Both Landau’s function g : N+ N+,n meo( ), and Chebyshev’s function ψ : N+
n
N+,n log(exp( )), are well-st→udied in a7→nalytic nSumber theory. Apart from information o→n
n
7→ S
their asymptotic growth behavior, explicit upper bounds are also available. More explicitly, Mas-
sias [8, Th´eor`eme, p. 271] proved that log(g(n)) 1.05314 nlog(n) for all n N+, and Rosser
and Schoenfeld [11, Theorem 12] that ψ(n)<1.0≤3883 n fo·rpall n N+. ∈
· ∈
The latter result translates into an exponential upper bound on Ψ := exp ψ. For n 27, the
◦ ≤
following best possible exponential bound on g(n) is sharper than the subexponential bound by
Massias, and its use will make some of our arguments easier:
Proposition 2.4.1. For all n N+, we have g(n) 3n3, with equality if and only if n=3.
∈ ≤
We conclude with the following consequence of Lemma 2.3.2:
Lemma 2.4.2. (1) Let G be a finite group, n N+. Then meo(G ) g(n) meo(Gn).
n
(2) Let S be a nonabelian finite simple group, ∈n N. Then g(n) m≀Seo(A≤ut(S)n·) < S n/3 implies
∈ · | |
that Λ(Aut(Sn))< Sn 1/3 and Λ (Aut(Sn))< Sn 2/3.
aff
| | | |
Proof. For (1): This follows immediately from Lemma 2.3.2(2).
For (2): Using Corollary 2.2.8(2), we conclude that Λ(Aut(Sn)) = meo(Aut(Sn)) = meo(Aut(S)
≀
) g(n) meo(Aut(S)n) < S n/3 = Sn 1/3, and that Λ (Aut(Sn)) = meo(Hol(Aut(Sn))) =
n aff
Smeo(≤Aut(Sn·)⋊Aut(Aut(Sn)))| |meo(A|ut(S| n)) meo(Aut(Aut(Sn)))=meo(Aut(Sn))2 < Sn 2/3.
≤ · | |
3 Reduction to the main lemma
The aforementioned “main lemma” is the following:
Lemma 3.1. Let G be a finite nonabelian characteristically simple group. Then:
(1) Λ(Aut(G))< G 13, with the following exceptions:
| |
8
Alexander Bors Cycle lengths and the solvable radical
(i) G = PSL (q) for some primary q 5. In this case, Λ(Aut(G)) = q +1, we have 1 <
∼ 2 ≥ 3
log (q+1) log(q+1) , and for q , this upper bound converges to 1 strictly monotonously
|G| ≤ log(1q(q2−1)) →∞ 3
2
from above.
(ii) G = PSL (p)2 for some prime p 5. In this case, Λ(Aut(G)) = p(p + 1), we have
∼ 2 ≥
1 < log (p(p + 1)) = log(p(p+1)) , and for p , this upper bound converges to 1 strictly
3 |G| log(1p(p2−1)) → ∞ 3
2
monotonously from above.
(iii) G∼=PSL2(p)3 for some prime p≥5. In this case, Λ(Aut(G))= 12p(p2−1)=|G|31.
(2) Λaff(Aut(G)) ≤ |G|23, with the following exceptions: G ∼= PSL2(p) for some prime p ≥ 5. In
this case, Λ (Aut(G)) = p(p+1), we have 2 < log (p(p+1)) = log(p(p+1)) , and for p ,
aff 3 |G| log(1p(p2−1)) → ∞
2
this upper bound converges to 2 strictly monotonously from above.
3
The purpose of this section is to show how to deduce all the main results from Lemma 3.1, so
until the end of this section, the word “proof” means “proof conditional on Lemma 3.1”. We first
give the precise definition of the constants E and E from Theorem 1.1.1:
1 2
Notation 3.2. (1) We set e :=log (6)=0.437618... and E := 1 = 1.778151....
1 60 1 e1−1 −
(2) We set e :=log (30) and E := 1 = 5.906890....
2 60 2 e2−1 −
Lemma3.3. (1)Forallfinitenonabelian characteristically simplegroupsG,wehaveΛ(Aut(G))
≤
|G|e1, with equality if and only if G∼=PSL2(5)∼=A5.
(2) For every ǫ>0, we have Λ(Aut(G)) G 13+ǫ for almost all finite nonabelian characteristically
≤| |
simple groups G.
(3) For all finitenonabelian characteristically simple groups G, we have Λaff(Aut(G)) Ge2, with
≤| |
equality if and only if G=PSL (5)= .
∼ 2 ∼A5
(4) For every ǫ>0, we have Λaff(Aut(G)) G 23+ǫ for almost all finite nonabelian characteristi-
≤| |
cally simple groups G.
Proof. The statements in (2) and (4) follow immediately from Lemma 3.1. For (1), note that by
Lemma 3.1(1), we have Λ(Aut(PSL2(5)))=6= PSL2(5)e1, and using the strict monotonicity of
| |
theupperboundsinLemma3.1(1),itisnotdifficulttoseethatthisistheonlycasewhereequality
holds. The proof of (2) is analogous.
Lemma 3.4. Let H be a finite semisimple group. Then:
(1) Λ(H) Soc(H)e1.
≤| |
(2) Λaff(H) Soc(H)e2.
≤| |
Proof. Let S ,...,S be pairwise nonisomorphic nonabelian finite simple groups, n ,...,n N+
1 r 1 r
such that Soc(H)=Sn1 Snr. Using the facts that Aut(H) embeds into Aut(Soc(H)),∈that
∼ 1 ×···× r
Λ(G) = meo(Aut(G)) for all finite semisimple groups G (Corollary 2.2.8(1,i)) and that Λ(R) =
meo(Aut(R))=Λ(Aut(R)) for allfinite centerlessCR-groupsR (Corollary2.2.8(1,i)and(2,i)), we
conclude that
Λ(H)=meo(Aut(H)) meo(Aut(Soc(H)))=Λ(Soc(H))=Λ(Sn1 Snr)
≤ 1 ×···× r ≤
Λ(Sn1) Λ(Snr)=Λ(Aut(Sn1)) Λ(Aut(Snr)) S e1n1 S e1nr = Soc(H)e1,
≤ 1 ··· r 1 ··· r ≤| 1| ···| r| | |
wherethe lastinequalityfollowsfromLemma3.3(1). Thisprovesthe inequalityin(1). For (2),we
usethefactthatH embedsintoAut(Soc(H)),thatΛ (G)=meo(Hol(G))forallfinitesemisimple
aff
groups G (Corollary 2.2.8(1,ii)) and that, by completeness of Aut(Sn1 Snr)= Aut(Sn1)
Aut(Snr), the tuple (Aut(Sn1),...,Aut(Snr)) has the splittin1g p×ro·p·e·r×ty,rto conclude,1wit×h
···× r 1 r
one application of Lemma 3.3(3) at the end, that
Λ (H)=meo(Hol(H)) meo(Hol(Aut(Soc(H))))=Λ (Aut(Soc(H)))=
aff aff
≤
=Λ (Aut(Sn1) Aut(Snr)) Λ (Aut(Sn1)) Λ (Aut(Snr))
aff 1 ×···× r ≤ aff 1 ··· aff r ≤
S e2n1 S e2nr = Soc(H)e2.
1 r
≤| | ···| | | |
9
Alexander Bors Cycle lengths and the solvable radical
Proof of Theorem 1.1.1. For (1), using the assumption as well as Lemmata 2.1.3(1) and 3.4(1),
we conclude that ρ λ(G) λaff(Rad(G)) λ(G/Rad(G)) 1 G/Rad(G)e1−1, and so [G :
≤ ≤ · ≤ ·| |
1
Rad(G)] ρe1−1. The proof for (2) is analogous.
≥
Proof of Corollary 1.1.2. The statements about cycle lengths in =PSL (5)follow immediately
A5 ∼ 2
from Lemma 3.1. As for the two asserted implications:
For(1): ByTheorem1.1.1(1)(andstrictmonotonicityofpowerfunctions), λ(G)> 1 implies that
10
[G:Rad(G)]<( 1 )E1 =60, and thus that [G:Rad(G)]=1.
10
For (2): This is similar to (1), but more involved. By Theorem 1.1.1(2), λ (G) > 1 implies that
aff 4
[G:Rad(G)]<(14)E2 =3600. So if any nonsolvable finite group G with λaff(G)> 41 existed, then
G/Rad(G)wouldhavesocleanonabelianfinitesimplegroupS oforderlessthan3600. ByLemma
2.1.3(2),itwouldfollowthatλ (S)> 1,soinordertogetacontradiction,itsufficestocheckthat
aff 4
λ (S) 1 for all nonabelian finite simple groups S such that S < 3600. By CFSG, there are
aff ≤ 4 | |
preciselyeight suchS,namely PSL (q) for q =5,7,9,8,11,13,17and . By Corollary2.2.8(1,ii),
2 7
A
itis sufficientto compute meo(Hol(S)) for these eightS, whichwedid withthe helpofGAP [3]. For
|S|
the PSL (q), the results are summarized in Table 1, and we also got that λ ( )= 1 :
2 aff A7 42
Table 1: λaff-values of the nonabelian finite simple groups of order smaller than 3600, excluding 7
A
q 5 7 9 8 11 13 17
1 1 1 1 1 1 1
λaff(PSL2(q)) 4 6 9 8 10 12 16
For proving Theorem 1.1.3, we introduce the following notation:
Notation 3.5. (1) For κ 0,2 and κ 0,1 , we denote by (κ) the set of finite nonabelian
∈ 3 aff ∈ 3 T
characteristically simple grou(cid:0)ps T(cid:3) such that Λ(cid:0)(Au(cid:3)t(T)) ≥ |T|31+κ, and by Ta(ffκaff) the set of finite
nonabelian characteristically simple groups T such that Λaff(Aut(T)) T 32+κaff. Note that by
≥ | |
Lemma 3.1, (κ) and (κaff) are finite.
T Taff
(2) For ǫ 0,2 , ǫ 0,1 and ρ (0,1), set
∈ 3 aff ∈ 3 ∈
(cid:0) (cid:3) (cid:0) (cid:3)
Λ(Aut(T)) Λ (Aut(T))
C(1)(ǫ,ρ):= and C(1)(ǫ ,ρ):= aff ,
Y T 31+ρǫ aff aff Y T 23+ρǫaff
T∈T(ρǫ) | | T∈T(ρǫaff) | |
aff
C(2)(ǫ,ρ):= T and C(2)(ǫ ,ρ):= T ,
| | aff aff | |
T∈TY(21ρǫ) T∈TY(21ρǫaff)
aff
C(ǫ,ρ):=C(1)(ǫ,ρ)ρ/12·ǫ ·C(2)(ǫ,ρ) and Caff(ǫaff,ρ):=Ca(1ff)(ǫaff,ρ)ρ/21·ǫaff ·Ca(2ff)(ǫaff,ρ),
D(ǫ,ρ):=max H +1 H a finite semisimple group such that Soc(H) <C(ǫ,ρ) ,
{| | | | | }
and
D (ǫ ,ρ):=max H +1 H a finite semisimple group such that Soc(H) <C (ǫ ,ρ) .
aff aff aff aff
{| | | | | }
Theorem 1.1.3(1) will follow rather easily from the following:
10