OKUYAMA, TAKASHI

Okuyama, Takashi

Osaka University Knowledge Archive

Finite groups admitting an automorphism of prime order fixing a cyclic 2-group

Osaka City University Repository

Okuyama, T.
Osaka J. Math.
18 (1981), 393-402
FINITE GROUPS ADMITTING AN AUTOMORPHISM
OF PRIME ORDER FIXING A CYCLIC 2-GROUP
TAKASHI OKUYAMA
(Received November 19, 1979)
1. Introduction
In this paper, we shall give a proof of the following Theorem, which is a
conjecture of B. Rickman [9]; in special case, CG(φ) has order 2, MJ. Collins
and B. Rickman proved in [2].
Theorem. Let G be a finite group which admits an automorphism φ of odd
prime order r whose fixed-point-subgroup CG(φ) is a cyclic 2-group. Then G is
solvable.
All groups considered in this paper are assumed finite. Our notation
corresponds to that of Gorenstein [7].
An important tool that is brought to attack the problem is B. Baumann's
classification of finite simple groups whose Sylow 2-subgroups are maximal
[1], and in analogy with Matsuyama [8] that used the results of [1], we shall
prove that HG(5;2)Φ1, where S is a φ-invariant Sylow 3-subgroup of G.
C.A. Rowley has obtained a proof of the theorem under the additional
hypothesis that G does not involve £4, the symmetric group on 4 letters.
The Theorem is a contribution to the continuing problem of showing
that finite groups which admit an automorphism φ of odd prime order such
that CG(φ) is a 2-group are solvable.
2. Preliminaries
We first quote some frequently used results.
2.1. (Thompson [12])
Let G be a group which admits a fixed-point-free automorphism of prime
order. Then G is nilpotent.
2.2. (Rowley [10])
Let G be a solvable group admitting an automorphism of odd prime order
p such that CG(φ), the fixed-point-subgroup of φ in G, is a cyclic (/-group,
Then, for any prime r, G is either r-nilpotent or r-closed.
394 T. OKUYAMA
2.3. (Glauberman [4])
Let G be a group with a Sylow ^-subgroup P, either p odd or p=2 and
£4 is not involved in G, in which CG(Z(P)) and NG(J(P)) both have normal ^>-
complements. Then G possesses a normal ^ -complement.
2.4. (Oilman and Gorenstein [3])
If G is a simple group with Sylow 2-subgrouρs of class 2, then G^L2(9),
q=7, 9 (mod 16), A7, Sz(2*)9 n odd, n>\, U3(2n), n^2, L3(2W), w^2, or P#
(4, 2«), ^^2.
2.5. (Gorenstein [7])
Let P be a Sylow ^-subgroup of G, where p is the smallest prime in π(G).
If p>2, assume d
n
(P)<z2, while if />=2, assume P is cyclic. Then G has a normal
P-complement.
2.6. (Matsuyama [8])
Let Q be a 2-group admitting an automorphism φ of odd order =t=l. If
d
c
(Q)= 1, then Q=E*R, where £ is φ-invariant, extra-special or 1, and R is
φ-invariant, and R is cyclic, D
m
, Q
my or Sm, w^4
2.7. (Collins-Rickman [2])
Let T be an extra-special 2-group admitting an automorphism φ of odd
prime order r acting fixed-point-freely on TjT'. Let S be the natural semi-
direct product 7Xφ> and let K be a field of nonzero characteristic different
from 2 and r. Assume that there exists a JCS-module M for which CM(φ)—
CM(Γ')=0.
Then (i) r=2w-fl is a Fermat prime,
(ii) \T\=22n+\ and
(iii) T^Q*(*D),
where Q and Z) denote the quaternion and dihedral groups of order 8, respec-
tively, and * denote the central product.
2.8. (Glauberman [5] [6])
Let G be a solvable group with a Sylow 2-subgroup Q with GΦC
(Z(Q))N(J(Q)), and O(X)=1. Put
Z=<Z*|G>Z*: 2-subgroup and O2(G/C(Z*))=1>
and/=<#<EG|#: 2-element, |Z/CZ(#)|=2>
and H= </, C(Z)>. Then the following hold
(i) there exists a normal subgroup G, of H containing C(Z), 1^/^gw, such
that, for i=l,...,ifi,Gf/C(Z)«S3, and JΪ/C(Z)=G1/C(Z)x - xGJC(Z).
(ii) let F,=[Gt ,Z], l^ί^w, and let Γ=F1Θ-ΘFIII, then
and F^Z2xZ
FINITE GROUPS ADMITTING AN AUTOMORPHISM 395
(iii) there is a 3 -element XQ of H such that, for each g^H, H=<(Q Π H, x0g,
C(Z)> and G/C(Z)=H/C(Z) CG/c(z)(xQ*C(Z)).
2.9. (Matsuyama [8])
Let G be a group with a Hall ττ-subgroup if, and let 1 φPeSy^/f), Q
2-group. If JVG(ff)-^ρ,^(Q)=l,Ω1(Z(Q))=<ec;>, C^)-l, and HG(P;;r')
= 1 , then, for each P8 ΦP, £ <Ξ G, m(P Π P*) ^  1 .
2.10. (Burnside's theorem [7])
If a Sylow ^-subgroup of G lies in the center of its normalizer in G, then G
has a normal ^ -complement.
2.11. (Burnside's theorem [7])
If P is a Sylow ^-subgroup of G, then two normal subsets of P are con-
jugate in G if and only if they are conjugate in NG(P). In paticular, two el-
ements of Z(P) are conjugate in G if and only if they are conjugate in NG(P).
2.12. (Smith-Tyrer [11])
Let G be a group with an Abelian Sylow ^-subgroup P for some odd prime
p. If [N(P):C(P)]= 2 and PΠN(P)' is noncyclic, then G is ^-solvable.
2.13. (Thompson Transitivity theorem [7])
Let G be a group in which the normalizer of every nonidentity ^-subgroup
is ^-constrained. Then if A^SCN3(P), CG(A) permutes transitively under
conjugation the set of all maximal ^ Z-invariant ^-subgroups of G for any prime q Φ
P-
2.14. (Collins-Rickman [2])
Let G be a group, and let p and q be distinct prime divisors of G.
Assume that G has an Abelian Sylow ^-subgroup P for which m(P)^3 and
that, whenever P0 is a subgroup of P with m(P/P0)^2, NG(P0) is ^ -constrained.
Then CG(P) permutes the elements of HG*(P;<?) transitively under conjugation.
2.15. (Frobenius theorem [7])
G is ^>-nilpotent if and only if NG(H)jCG(H) is a ^>-group for every noniden-
tity ^-subgroup H of G.
3. The proof of the Theorem
Let G be a minimal counterexample to the Theorem, for the remainder of
this paper.
Lemma 3.1. G is simple.
Proof. By Lemma 5.1. of [2],
396 T. OKUYAMA
Lemma 3.2. Let p be a prime divisor of G and P^Sylp(G). If NG(P)
has a normal p-complement, then p=2 and the symmetric group S4 is involved in G.
Proof. By Lemma 5.2. of [2], (2.2) and (3.1).
For the remainder of this paper, Q denotes the φ-invariant Sylow 2-
subgroup of G, and let CG(φ)—<#> and Ωι(CG(φ))=<«;>.
Then Q is a unique φ-invariant Sylow 2-subgroup, and let p be an odd
prime in π(G) and PeSly,(G), then, by (3.2), NG(P)=>w.
Lemma 3.3. d
c
(Q)^2.
Proof. If d
c
(Q)=l, by (2.6) and hypothesis Q=E*R where E is φ-
invariant, extra-special, and R is φ-invariant, cyclic. If E=l, by (2.5) G is
2-nilρotent, contrary to (3.1). So ί'Φl. Since df(Q)=2, by (2.4) this is a
contradiction.
Lemma 3.4. Every φ-invariant proper subgroup of G is 2-nilpotent.
Proof. Assume otherwise. Let M be a non-nilpotent maximal φ-invariant
subgroup of G without a normal Sylow 2-subgrouρ. If N(O2(M)) is 2-nilpotent,
M is nilpotent, a contradiction. By (2.2), N(O2(M)) is 2-closed. Hence M=
N(O2(M}\ O2(M)=0, and M=NG(Q). Thus there is an odd prime p dividing
the index [NG(Q):CG(Q)].
By (3.3), there is a characteristic subgroup C of Q such that C^Z2X ••• XZ2,
C contains £1^ (0)), and [C, φ] = l. Let P0 be a φ-invariant Sylow p-
subgroup of NG(Q) and P be a φ-invariant Sylow ^-subgroup containing P0.
We now claim that [C, P0] = 1. We may assume that w e C. [zυ, P0] C Q Π
P=l. Since P0 centralizes C/CC(P0), [P0, C] = l. Thus CCNG(P0).
Let MO be a maximal φ-invariant subgroup containing NG(P0). If M0 is
2-closed, M0=JVG(0). Since NP(P0)=P0, P=P0. Let Q0 be a φ-invariant
Sylow 2-subgroup of NG(P). Then [P, Q0]£P Π 0=1, so ^VG(P) is P-nilpotent,
and by (3.2), p=2, a contradiction. Thus M0 is 2-nilpotent. Hence MQ=
NG(P). Since CC7VG(P), 1 φ[C, φ]CCc(P).
Now put Z0=[β1(Z(0)), φ] If Z0=l, P,ρCCG(Z0). When CG(Z0) is
2-closed, PCΛ^G(0), and [00, P0]£βnP=l, a contradiction. Hence CG(Z0) is
2-nilpoent. Therefore as QS>NG(P), [^,P0]£QnP=l, a contradiction. Thus
we may assume that Z0— 1, hence that Ω,l(Z(Q))=^u)y.
Put Q=Q/<wy and let Cl be the inverse image of Z(0) Π C in Q. As [C^^]
C<», CiCTVc^)). On the other hand, let jyeCV Then [y,φ]eCG«Λ»,
since (y"1Λ?y)*=j; 1Λ;y. Put C0— [Cj,φ], so that 1 φC0£ΛΓG(P), hence CG(P0)
contains P0 and x.
Now let M1 be a maximal φ-invariant subgroup of G containing CG(C0). If
M! is 2-closed, M1=NG(Q), and [00) -P]=1, contradiction. Thus M1 is 2-nilpotent,
FINITE GROUPS ADMITTING AN ATUOMORPHISM 397
i.e. Mi=NG(P).
Put Q=Q/Φ(Q). |>,P0]CPΓlQ=l. Since P0 centralizes 0/Cg(P0), P0
centralizes Q. Hence [P0,Q]=ly a contradiction. Hence the lemma is proved.
For the remainder of this paper, in analogy with Matsuyama [8], we shall
prove the following result;
(0 3 / | G | ;
(ii) I CG(S) I is odd, where S is a φ-invariant Sylow 3-subgroup of G;
(iii) HG(5;2)φl;and
(iv) m(S)^4.
On the other hand, in analogy with Collins- Rickman [2], we shall prove that
HG(S 2) — 1 . Hence this contradicts above.
For the remainder of this paper, we shall write down the results which can
be similarly proved as [8].
(3.5) CG(w)ZQ.
(3.6) If p is an odd prime in τt(G) and PeSyl^G), then P is Abelian.
(3.7) If p is an odd prime in π(G) and A is any ^ -subgroup of G, then
AutG(A)=NG(A)ICG(A) is a 2-group.
(3.8) If Ωi(Z(Q)) Φ<», then NG(T) is a 2-group for any nontrivial φ-
invariant 2-subgroup T of G.
Now put P be a φ-invariant Sylow ^-subgroup of G for any odd prime p in
π(G). Let Kp be a normal 2-complement of NG(P) and Qp= Q Π NG(P). Then
NG(P)=QpKpί Q,£Q. Furthermore let Q*=CQf(Kp), and then Q*=[Q*, φ],
since w&Q*.
Hence, for any s^π(K
s
), Kp=Ks, Qp=Qsy and Q^*=Qβ*. In particular,
Kp is a nilpotent Hall subgroup of G.
(3.9) CQp(P)=Q*.
(3.10) d
c
(Qp/Q*)=l.
Furthermore let_Mp=NG(P) and_Mp=Mp/Q*Kp. Then by (2.6) and
hypothesis, Mp=Ep*Ep, where either Ep—l or Ep is φ-invariant, extra-special
and Rp is φ-invariant, cyclic.
On the other hand, by (3.4), NG(Q) is nilpotent, and then NG(Q)=Q by (3.5).
Hence by (3.2), S4 is involved in G, yields 3/ 1 G | . Furthermore let SeSyl3(G),
and then
Lemma 3.11. Let p be an odd prime in π(G). We can write Mp=Ep*ϊfpy
where either Ep=l or Ep is φ-ίnvarίant, extra-special, and Rp is ^-invariant,
398 T. OKUYAMA
cyclic.
If SjΦl, then r=2n+ί is a Fermat prime.
Proof. By (2.7), it is immediate that CQί(P)(φ)=CQl(P)(Epr)=Q. By
(2.7), it suffices to prove that φ acts on Ep/Ep' fixed-point-freely. First we may
assume that | Rp \ =2. Then, since we can suppose that φ centralizes an element
of Ep of order 4, it is not necessarily trivial.
Now suppose that there exists an element y of Ep of order 4 such that [y,
φ]=l. As Ep is extra-special, the conjugate class of y is {y, yw}. Hence
[Ep: C-Ep(y)]=2. Then φ acts on the set, Ep—C%p(y), fixed-point-freely. It
is impossible.
Lemma 3.12. Let S be a φ-invariant Sylow 3-subgroup of G. If [ζ?3/Q3*, φ]
= 1, then Sis a T.I.-set.
Proof. If not, there exists an element g of G such that S8^pS and Sg Π S Φ1.
First we shall show that Cρ3(#) = Q3* for any #eS*. It is immediate that
Cg3(*)2Q3*. If Cζ>3(*)SQ3* for some z<=S*, w^CQa(z), by hypothesis. But
this is impossible. Next we will prove that, for any #eS*, CG(z) is 3-nilpotent.
Now put CG(z) = C and let S1 be a nontrivial subgroup of S. By (3.7),
Aut^Sj) is a 2-grouρ. Put Aut
c
(ιSΊ)3ί Φl. Then t is a 2-element. Furthermore
there exists an element y of S
λ
 such that y* Φy, i.e. y and y* are conjugate in
CG(z). By (2.11), y and yt are conjugate in NC(S). Thus we may assume
that t(=N
c
(S), and ίeQ3. Then t<=CQ3(z)=Q* = CQ3(S), a contradiction.
Hence CG(z) is 3-nilρotent by (2.15), especially CG(z) is 3-constrained.
Furthermore put 3Φp^π(K3), and let P be a φ-invariant Sylow ^-sub-
group of G. NG(S)=NG(P). Thus CG(z) is τr(^3)-nilpotent.
Next put 1Φ3/<ΞS*Γ!S, and let M be a τr(^3)-complement of CG(y), and
then we will prove that M is a 2-group.
S normalizes M and ( |*SΊ, \M\)=1. Now suppose that M is not a 2-
group. There exists an odd prime q in π(M) such that qξ£π(K3). Furthermore
there exists a Sylow (/-subgroup Q1 of M normalized by S. Since Aut^Q^ is
a 2-group, /SCC^Qj), and hence Q^K3. It is impossible. Thus M is a 2-
group.
On the other hand, it is easy to show that MΞ>Q3*. Now suppose that M=
Q3*. Then CG(y)=Q3* K3, and since S, S8S>CG(y), S=Sg, a contradiction.
Hence M3Q3*, CG(S)^NG(M).
Let M be the intersection of all elements of HG*(5;2). By (2.14), MSM.
On the other hand, as M is φ-invariant, SM is φ-invariant. By (3.4), SM is 2-
nilpotent. Thus [M,S]£MΓϊ*S=l. Hence MCCG(S)=Q3*, a contradiction.
This completes the proof of Lemma 3.12.
Now if Z?3φl, r=2
n+l is a Fermat prime by (3.11), where r=\φ\.
FINITE GROUPS ADMITTING AN ATUOMORPHISM 399
On the other hand, when E3=ίy by (3.12), S is a T.I.-set, where S is a
φ-invariant Sylow 3 -subgroup of G.
By B. Baumann [1], Q is not a maximal subgroup of Gy and thus there
exists a proper subgroup X of G containing Q such that Q is a maximal sub-
group of X.
In analogy with Matsuyama [8], we can say the following.
X is a solvable {2, 3} -subgroup with O(X) = l, and X satisfies the hypo-
thesis of (2.8). Thus the structure of X is one of the following two type.
<Type I>
X/O2(X) is isomorphic to *S3, the symmetric group on 4 letters.
Z(02(X)) contains Z(Q) and Z(O£X)) = [Z(O£X))9X]®CzWx»(X),
where [Z(O2(X)),X] is isomorphic to Z2XZ2.
<Type II>
X has a subgroup H containing O2(X) such that [X: H] = 2. HjO2(X)
=X1IO2(X)xX2(O2(X)),XilO2(X) is isomorphic to 53,*= 1,2. Z(O2(X))
contains Z(Q) and Z(O2(*))=[Z(O2(*)), XA®[Z(OJX)),XA®Czl0tlx»
(H), where [Z(O2(J?)), -SΓ. ] is isomorphic to Z2χZ2, i=l,2.
On the other hand, considering the structure of X, Z(Q) is noncyclic, by
(3.8), <?,*=!.
Now we will show that HG(J^3;2)Φ1. For the remainder of this paper,
let S be a φ-invariant Sylow 3 -subgroup of G.
Lemma 3.13. ΊΛG(S;π(K3y)=iaG(S 92).
Proof. It is easy that HG(S;π(K3)')^ΉG(S',2). If there exists an element
A of HG(S;π(K3)') that is not a 2-grouρ, by [7; 6.2.2], S normalizes some Sylow
^-subgroup 5* of A. As AutG(5'*) is a 2-grouρ, [5,5*] = 1. But it contra-
dicts CG(S)=K3.
By (3.13), it sufEces to prove that HG(S;
Now we suppose that HG(S',π(K3)')= 1. By Matsuyama [8], we can say
the following.
(3.14) If S* ΦS, £6ΞG, then m(S Π S*)^l.
(3.15) There exists a nontrivial proper subgroup Z
λ
 of Z(Q) such that
3/|C
β
(Z1)|and[Z(Θ):ZJ=2.
Furthermore, in analogy with Matsuyama [8], we can show the next lemma.
Lemma 3.16. There exists a nontrivial element a of Ωi(Z(ζ))) such that
> i- 1 0<*> I or
400 T. OKUYAMA
Proof. Put α^eΩ^ZίQ))1, a^w. Let Al=^{a^>}. If there exists an
element of Ω1(Z(Q))*— ^ 4X that does not equal w, let a2 denote this element. So
let A2={a2<φ>$, and then A1Γ\A2= φ. Inductively, if there exists an element
of Ω!(Z(Q))*— \]Ai that does not equal «;, we let a{ denote this element. Then
we can write the following,
Ω1(Z(Q))»-<w> =
ι=l
where -4f n-4y=φ if ίφj
Now suppose that m^2. Let |Ωi(Z(Q))| =2", and as
=2, I Ω^Zi) I =2»"1. If, any i, l^i^m, \ a^> Π Z
x
 | ^ - - 1 αf <φ> | , then since
I *,-<*> I =r is odd. I \](a^> Π ZJ | ^  | ^ (Z(Q))-^(ZO | -2.
But, on the other hand, lΩ^Z^-Ω^ZOI -2«-1, and |Ω1(Z1)I|=2I|-1-1.
It is impossible. Hence m=l. Ωi(Z(Q))*= {α }^ . This lemma is proved.
(3.17) aφt normalizes some Sylow 3 -subgroup of G, O^z^r — 1.
Now put Δ, =(tfφ<)GnQ3, and then Δ. Φφ, and Δf φ=Δί+ι, O^i^r— 1.
Furthermore, as Q3*=l, Q3=E3*R3.
If ^^l, then 5 is a T.I. -set, by (3.13). In analogy with the above ar-
gument, we can show that Δt Φφ, 0<C/O — 1.
But, in this time, w is an only involution in Q3. This is a contradiction.
Hence, for the remainder of this paper, we may assume that /?3Φ1, i.e. r=
2n+l is a Fermat prime. Then (3.16) is reduced that there exists a nontrivial
element a of ^(Z(Q}) such that
On the other hand,
(3.18) There exists an element bi9 bj of Δ, ,Δy, respectively, O^/^f^r— 1,
*>;,[*,, *>]=!.
Next Δf is determined as the following.
Lemma 3.19. Δf = {b{, b^}, O^i^r— 1, b{ *w.
Proof. If weΔ, , then ίί; centralizes some element of order 3, a contradic-
tion.
For the remainder, we set ό—έ,-.
Suppose that 6, fi'eΔf, ^eG, b φK Then 6, έ^eQ3. Since S=
s
Let S* be a Sylow 3-subgroup of CG(bg) containing Cs(bg). There exists
an element h of CG(b*) such that (Cs(i))'*£5*. On the other hand, let SQ be a
FINITE GROUPS ADMITTING AN AUTOMORPHISM 401
Sylow 3-subgrouρ of G containing S*, and then S=S0 as Cs(bg)^S Π 50. Since
(C5(i))**C S Π S'*, £* £ΞNG(S). Since i*=δ**, έ and V are conjugate in NG(S).
As NG(S)—Q3K3, b and i* are conjugate in Q3. Hence Δ^ {bh
Now put Δ=<Δf|0^ι'ί£r— 1>, and then, by (3.19), Δ is φ-invariant
Abelian. Furthermore, as [Δ, φ]φl, [Δ,φ]jSΓ3 is nilpotent, and
this is a contradiction. Hence HG(5;2) Φl.
On the other hand, we will prove the next lemma, and then, in analogy
with Collins- Rickman [2], the proof of the main theorem is complete.
Lemma 3.20. Let S0 be a proper subgroup of S such that m(S/S0)^2.
Then NG(S0) is ^solvable.
Proof. First we shall consider the case m(S0)>2. In this case, we will
show that CG(S0) is 3-nilpotent. Put C=CG(50), and let Sl be a nontrivial
subgroup of S. If there exists a nontrivial element t of Aut
c
(S1), t is a 2-element
as Aut
c
(SΊ) is a 2-group. Then there exists an element y of S1 such that y* Φ^y.
Thus y and yl are conjugate in CG(S0). By (2.11), y and yt are conjugate in
NC(S). Hence we may assume that t<=Q3Γ\C^Z2X ••• xZ2. As ίφw, A?=
Hence
±m(S) = m(C
s
(t)) = m(C
s
(tzv)) .
This is a contradiction. By (2.15), CG(S0) is 3-nilpotent. CG(S0)IS0 is
3-solvable. Hence NG(S0) is 3-solvable.
Now we may assume that m(S)=4 and m(S0)=2. In this case, similarly,
if CM9(S0) = C(S), M3=NG(S), then by (2.10), CG(S0) is 3-nilpotent. Hence,
furthermore, we may assume that CM3(50) 3(7(5).
If there exists an element x0 of CM3(S0) such that |Λ? O |= 4, then xQ2=w^
CM3(S0), a contradiction.
If there exists a four-group <^ι>X<^2> in CM3(*S0), then Sf=<Cs(jc1), Cs(^2)j
C
s
(xlx^y. On the other hand, *S0 is contained in Cs(x^ Cs(x2), and CS(^Λ)>
and since
™(C
s
(xι)) = m(C
s
(x2)) = m(Cs(x1xz)) = 2 ,
C
s
(x^=C
s
(x2)=Cs(xlx2)=S09 a contradiction. Hence we can write the follow-
ing;
where t2<=ΞC(S) and S=S00 [£,*].
402 T. OKUYAMA
Put CG(ιSro)=C'G(5ro)/ιS0. Then S=SnN(c^(S)'. By (2.12), CG(S0) is
3-solvable. Hence, in this case, NG(S0) is 3-solvable. This lemma is complete.
Now we already proved that HG*(S;2)Φ1. Next we will show that there
exists a φ-invariant element Q
λ
 of HG*(*S;2). Suppose false. Since HG*(S;2)
is φ-invariant, r divides |HG*(*S';2) |. On the other hand, by (2.14), the element
of HG*(S;2) permuted by C(S) transitively. This is a contradiction.
Let N=SQ1. By (3.4), N is nilpotent. Hence
On the other hand, as Q3*=l, \CG(S)\ is odd. This is a contradiction.
The main theorem is proved.
Acknowledgement.
The author thanks to Mr. Fukushima and Mr. Matsuyama for this theme
and for their helpful advices.
Reference
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Sayama Industrial High School
Sayama-shi, Fujimi 2-5-1
Saitama-ken 350-13
Japan


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Finite groups admitting an automorphism of prime order fixing a cyclic 2-group

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