Journal of Algebra 240, 665᎐679 Ž2001.
doi:10.1006rjabr.2001.8740, available online at http:rrwww.idealibrary.com on
On Quasi-thin Association Schemes with Odd Number
of Points
Mitsugu Hirasaka1
Combinatorial and Computational Mathematics Center, Pohang Uni¨ ersity of Science
and Technology, Pohang 790-784, Korea
E-mail: [email protected]
Communicated by Walter Feit
Received February 23, 2000
Let Ž X, R . be an association scheme in the sense of P.-H. Zieschang Ž1996, ‘‘An
Algebraic Approach to Association Schemes,’’ Lecture Notes in Mathematics, Vol.
1628, Springer, New YorkrBerlin., where X is a finite set and R is a partition of
X = X. We say that Ž X, R . is quasi-thin if each element of R has a valency of at
most two. In this paper we focus on quasi-thin association schemes with an odd
number of points and obtain that Ž X, R . has a regular automorphism group when
n O ␪ Ž R . is square-free.
䊚 2001 Academic Press
1. INTRODUCTION
Let Ž X, R . be an association scheme Žor simply, a scheme. in the sense
of w9x where X is a finite set and R is a partition of X = X. It is
well-known that each transitive permutation group G, say, of SymŽ X .
forms a scheme as the orbitals of G Žsee w2x., where SymŽ X . is the set of
all permutations of X and an orbital is an orbit of the induced action of G
on X = X. However, each scheme could not be realized as the orbitals of
a transitive permutation group. The purpose of this paper is to find a
sufficient condition for the existence of a transitive automorphism group.
We say that an element of R is a relation. For each relation r, say, the
pair Ž X, r . is a digraph whose out-degree function is a constant, called the
valency of r. We say that a scheme is thin Ž quasi-thin. if each relation has
a valency of at most one Žrespectively, two..
1
The author thanks the Combinatorial and Computational Mathematics Center of Pohang
University of Science and Technology for their support.
665
0021-8693r01 $35.00
Copyright 䊚 2001 by Academic Press
All rights of reproduction in any form reserved.
666
MITSUGU HIRASAKA
We will begin by considering thin schemes. In w9, p. 177x it is shown that
each thin scheme is realized as the orbitals of the regular permutation
group of a group. It is a natural direction to consider quasi-thin schemes as
the second step, although this is much more variant than for thin schemes.
In contrast to a relation of valency one, there is no way to classify a
relation of valency two unless the relation is symmetric, which is one of the
causes for the variety. Let us focus on a relation r, say, of valency two in
not only quasi-thin schemes but also in general schemes. There have been
some attempts to investigate or characterize Ž X, r . under various assumptions Žsee w1, 5, and 7x.. In w7x orbitals of half-transitive actions are
examples of nonsymmetric relations of valency two. In w1x all of the
intersection numbers of Ž X, R . are classified if Ž X, r . is connected and
there is no nondiagonal relation of valency one. In w5x the world of
quasi-thin schemes is introduced as a special class of schemes with a
relation of valency two. In a sense this paper is a branch of w5x.
Let us summarize the main points obtained in w5x Žcf. Proposition 2.2.: If
Ž X, R . is quasi-thin, then there exists a subgroup G, say, of AutŽ X, R .
whose orbitals coincide with the equivalence relation induced by O ␪ Ž R .
Žsee Section 2 for the definition.. Thus, we have substructures induced by a
␪
permutation group, all of which are ranged in O ␪ Ž R .. Since Ž X, R . O Ž R.
␪
Žsee Section 2 for the definition. is thin, the group induced by Ž X, R . O Ž R.
and G gives much influence to the whole structure. What we need to do in
order to find a transitive automorphism group is to establish the relationship between the orbitals of G and O ␪ Ž R . Žsee Section 3.4. and to lift it up
to the whole Žsee Lemma 3.2.. Thus, we see that a quasi-thin scheme is an
object which is strongly influenced by group actions. The following is a
standard construction of quasi-thin schemes.
EXAMPLE 1.1. Let G be a finite group which contains t with t 2 s id G .
Then G acts on the right cosets Gr² t : by right multiplication. Since the
above action is transitive and <² t : g ² t :<r<² t :< F 2 for each g g G, the
orbitals of G form a quasi-thin scheme.
In this paper we purpose to prove that each quasi-thin scheme such that
the valency of its thin residue is square-free can be realized by way of
Example 1.1. The reason for the restriction on the valency of the thin
residue comes from the existence of intransitive quasi-thin schemes where
we say that a scheme is transiti¨ e Žrespectively, intransiti¨ e . if its automorphism group is transitive Žrespectively, intransitive . on X. Hanaki and
Miyamoto found an intransitive quasi-thin scheme whose thin residue has
valency four, listed as No. 176 in w4x.
Let us now give the organization of this paper. We give terminology and
some basic results in Section 2. In Section 3 we define the arranged maps.
In Section 4 we focus on quasi-thin schemes and prepare two propositions
QUASI-THIN ASSOCIATION SCHEMES
667
for the proof of our main result, Theorem 5.1. In Section 5 we obtain
Theorem 5.1 and Corollary 5.2 as an application of Proposition 4.7. We
conclude from Corollary 5.2 that all relations of quasi-thin schemes with
the assumption given in Corollary 5.2 could be characterized as a Cayley
graph.
2. TERMINOLOGY AND BASIC RESULTS
Following w9x we give the notation about association schemes. Let X be
a finite set. Given r ; X = X and z g X, we set
1 X [ Ž x, x . N x g X 4 ,
r* [ Ž x, y . N Ž y, x . g r 4 ,
zr [ y g X N Ž z, y . g r 4 .
Let R be a partition of X = X which does not contain the empty set. We
say that Ž X, R . is an association scheme Žor simply, a scheme. if it satisfies
the following conditions:
Ži.
Žii.
Žiii.
on d, e,
1 X g R;
for each r g R we have r* g R;
for all d, e, f g R and each Ž x, y . g f, < xd l ye* < depends only
f where we denote the cardinality of any finite set ⍀ by < ⍀ <.
We denote < xd l ye* < by a d e f , and a d e f N d, e, f g R4 are called the
intersection numbers of R. For each r g R we abbreviate n r [ a r r*1 X ,
which is called the ¨ alency of r. For each Ž x, r . g X = X we denote the
unique element of R which contains Ž x, y . by r Ž x, y ..
For each F : R and each x g X we set
nF [
Ý nf
xF [
and
fgF
D xf .
fgF
Following w9x, we define the complex product 2 of E and F as
½
EF [ r g R
Ý Ý ae f r / 0
egE fgF
5
for all E, F : R.
For convenience we shall write eF and Ef instead of e4 F and E f 4 ,
respectively, where e, f g R.
2
It is a trivial observation that the complex product is an associative operation.
668
MITSUGU HIRASAKA
A subset F : R is called closed if FF : F.3 We shall denote by C Ž R .
the set of all closed subsets of R. For each E : R we set
² E: [
F F g C Ž R. N E : F 4 .
We shall write E F F if E : F and E, F g C Ž R ..
Following w9x, for each F g C Ž R . and x g X we set
Ž X , R . x F [ Ž xF , f x F 4 fgF . ,
f x F [ f l Ž xF = xF . .
Then Ž X, R . x F is an association scheme, which is called the subscheme of
Ž X, R . with respect to Ž F, x .. We set
XrF [ xF N x g X 4
and
RrrF [ r F N r g R 4 ,
where r F [ Ž yF, zF . N z g yFrF 4 . Then Ž X, R . F [ Ž XrF, RrrF . is an
association scheme, which is called the factor scheme of Ž X, R . over F.
The intersection numbers of Ž X, R . F may be computed by the formulae
Žsee w9, p. 21x.
Ž 1.
ad F e F f F s
1
nF
Ý
Ý
a b c f , in particular n r F s
bgFdF cgFeF
nF rF
nF
,
and
Ž 2.
n F < XrF < s < X < .
We say that F : R is thin Ž quasi-thin. if n F s 1 Žresp., n f F 2. for
f g F. For each F g C Ž R . we set
O␪ Ž F . [ f g F N n f s 1 4
and
O␪ Ž F . [
¦D ;
ff * ,
fgF
which are called the thin radical and the thin residue of F, respectively.
We say that F g C Ž R . is symmetric if f * s f for each f g F and
commutati¨ e if a d e f s a e d f for all d, e, f g F. It is well-known that each
symmetric closed subset is commutative.
We shall write the automorphism group of Ž X, R . as
Aut Ž X , R . [ ␴ g Sym Ž X . N ᭙ w, z g X , r Ž w, z . s r Ž w ␴ , z ␴ . 4 .
3
If < X < is finite, then FF : F is equivalent to FF* : F where F* [ Df g F f *.
QUASI-THIN ASSOCIATION SCHEMES
669
LEMMA 2.1 w1, Prop. 5.1; 9x. Let Ž X, R . be an association scheme. For
all d, e, f g R we ha¨ e the following:
Ži. n d n e s Ý f g R a d e f n f ;
Žii. a d e f n f s a f e*d n d s a d*f e n e ;
Žiii. a d1 e s ␦ d, e , where ␦ d, e is the Kronecker’s delta;
X
Živ. gcdŽ n d , n e . G < de <.
PROPOSITION 2.2 w5x. Let Ž X, R . be a quasi-thin scheme. We set G [
² ␾ x N x g X : where ␾ x g SymŽ X . such that ␾ x Ž y . [ y⬘, xr Ž x, y . s y, y⬘4
for each y g X.4 Then we ha¨ e the following:
Ži. G F AutŽ X, R .;
Žii. the orbits of G on X coincide with XrO ␪ Ž R .; and
Žiii. if O ␪ Ž R . ­ O␪ Ž R ., then G acts faithfully and transiti¨ ely on
␪Ž .
xO R and ŽAutŽ X, R .. x s ² ␾ x : where ŽAutŽ X, R .. x is the stabilizer of x in
AutŽ X, R ..
3. ARRANGED MAPS
We assume that Ž X, R . is an association scheme and write 1 [ 1 X for
the remainder of this paper.
Let F g C Ž R . and x, y g X. A map ␴ : xF ª yF is called arranged
with respect to Ž F, x, y . if it satisfies the following conditions:
Ži. ␴ is a bijection with ␴ Ž x . s y;
Žii. for all w, z g xF we have that r Ž ␴ Ž w ., ␴ Ž z .. s r Ž w, z ..
We say that F g C Ž R . is arranged if for all x, y g X there exists an
arranged map with respect to Ž F, x, y ..
LEMMA 3.1. The following are equi¨ alent:
Ži. Ž X, R . is transiti¨ e;
Žii. R is arranged;
Žiii. each F g C Ž R . is arranged.
Proof. Ži. « Žii.. Let x, y g X be two arbitrary points. Then there
exists ␴ g AutŽ X, R . such that ␴ Ž x . s y. From the definition of AutŽ X, R .
it is clear that ␴ is an arranged map with respect to Ž R, x, y ..
Žii. « Žiii.. Since R is arranged, there exists an arranged map ␶ with
respect to Ž R, x, y .. Since the restriction of ␶ on xF is an arranged map
with respect to Ž F, x, y ., we conclude that F is arranged.
4
Note that xr Ž x, y . is the neighborhood of x with respect to r Ž x, y ..
670
MITSUGU HIRASAKA
Žiii. « Ži.. Since an arranged map with respect to Ž R, x, y . is an
automorphism of Ž X, R ., we conclude that AutŽ X, R . is transitive on X.
LEMMA 3.2 w6x.5 Let E, T g C Ž R . such that T F O␪ Ž R . and ET g C Ž R ..
If E is arranged, then ET is arranged. In particular, each thin closed subset is
arranged.
Proof. Let x, y g X be two arbitrary points and T be a right transversal of E l T in T, i.e., T s D
qt g T Ž E l T . t. Without loss of generality we
may assume that 1 g T.
We claim that each z g xET has a unique presentation in a z s ˜
zt
T. In order to
where ˜
z g xE and t g T. Since T s Ž E l T . T , xET s xET
prove the uniqueness, it suffices to show that if us s ¨ t with u, ¨ g xE
and s, t g T , then u s ¨ and s s t. Since ust* s ¨ 4 and u, ¨ g xE,
st* g E l T and, hence, s g Ž E l T . t. This implies that t s s and, hence,
u s ¨.
Since E is arranged, there exists an arranged map ␴ with respect to
Ž E, x, y .. Define a map ␶ : xET ª yET such that ␶ Ž wt . [ ␴ Ž w . t for each
w g xE and t g T. We claim that ␶ is arranged with respect to Ž ET, x, y ..
It follows from the previous paragraph that ␶ is a bijection. By the
definition of ␶ , ␶ Ž x . s ␶ Ž x1. s ␴ Ž x .1 s y. For all ws,
˜ ˜zt g xET with
w,
˜ ˜z g xE and s, t g T we have
r Ž ␶ Ž ws
˜ . , ␶ Ž ˜zt . . s r Ž ␴ Ž w
˜ . s, ␴ Ž ˜z . t .
s rŽ ␴ Žw
˜ . s, ␴ Ž w
˜. . r Ž ␴ Ž w
˜ . , ␴ Ž ˜z . . r Ž ␴ Ž ˜z . , ␴ Ž ˜z . t .
s s*r Ž w,
˜ ˜z . t s r Ž ws,
˜ ˜zt . .
Therefore, ␶ is arranged with respect to Ž F, x, y ..
The second statement is obtained by setting E s 14 .
4. GROUP-ARRANGED MAPS
LEMMA 4.1. For each element f g R with n f F 2 there exists a unique
element, denoted by s f , such that ff * [ 1, s f 4 . Moreo¨ er, Ž s f .* s s f .
Proof. It is obvious that if n f s 1 then s f s 1. If n f s 2, then, by
Lemma 2.1Živ., < ff * < F 2. Since 1 g ff *, the first statement follows from
Lemma 2.1Ži, iii.. From the definition of the complex product, r Ž x, y . s s f
if and only if there exists z g xf l yf and x / y. Therefore, the second
statement follows.
5
Although this lemma was stated under the assumption that T is cyclic before, it was
improved to the present style in w6x. We shall give a proof here for the reason that w6x is in a
state of preparation.
QUASI-THIN ASSOCIATION SCHEMES
671
We assume that F g C Ž R . is quasi-thin for the remainder of this
section.
LEMMA 4.2.
If O␪ Ž F . s 14 , then we ha¨ e the following:
Ži. n F is odd;
Žii. F is symmetric;
Žiii. O ␪ Ž F . s F.
Proof. Ži. Since n F s Ý f g F n f and O␪ Ž F . s 14 , n F should be odd.
Žii. Assume the contrary, i.e., that there exists f g F with f * / f
and n f s 2. Then ² f : has a nontrivial thin element Žsee w1, Theorem 1.3x.,
a contradiction.
Žiii. Let f g F. Since f is symmetric by Žii. and 2 ¦ n ² f : by Ž2.,
²
f g s f : and, hence, f g ² s f : F ²Df g F ff *: s O ␪ Ž F .. Thus, O ␪ Ž F . s F.
In this section we focus on the case where n O ␪ Ž F . is odd. Lemma 3.2
shows that each thin closed subset is arranged. Therefore, for the remainder of this section we assume that n O ␪ Ž F . is odd and greater than one, so
that F is not thin.
For each x g X the subscheme with respect to Ž F, x . is quasi-thin. Since
2 ¦ n F and n O ␪ Ž F . ) 1, we conclude from Lemma 4.1 that there exists
f g F such that s f f O␪ Ž F .; in particular, O ␪ Ž R . ­ O␪ Ž R .. Applying
Proposition 2.2 for Ž X, R . x F , we obtain that
G Ž x . [ ² ␾ z N z g xF : F Aut Ž X , R . x F
and a point-stabilizer in G Ž x . is of order two where ␾ z : xF ª xF is as
given in Proposition 2.2. Since < GŽ x .< s 2 < x GŽ x . < s 2 n O ␪ Ž F . and 2 ¦ n O ␪ Ž F .
by Ž2., GŽ x . has a unique subgroup H Ž x ., say, of index two Žsee w8,
Theorem 4.5x.. Furthermore, H Ž x . is regular on x GŽ x ., since 2 ¦ < H Ž x .< s
< x GŽ x . < and H Ž x . x F G Ž x . x s ² ␾ x :.
For each E F O ␪ Ž R . we denote by H Ž E, x . the setwise-stabilizer of xE
in H Ž x .. It follows from w8, Theorem 7.4x that H Ž E, x . is invariant by the
conjugation of ␾ x ; conversely, for each K F H Ž x . with K ␾ x s K we have
C Ž K . [ r Ž x, x k . N k g K 4 F O ␪ Ž F ..
LEMMA 4.3.
following:
Ži.
Žii.
Žiii.
For each x g X and for all h, k g H Ž x . we ha¨ e the
␾
x
r Ž x, x h . s r Ž x, x h . s r Ž x k , x h k .;
if r Ž x, x h . s r Ž x, x k ., then k g h, h␾ x 4 ;
␾x
r Ž x, x h . r Ž x, x k . s r Ž x, x k h ., r Ž x, x k h .4 .
672
MITSUGU HIRASAKA
Proof. For convenience we set ␾ [ ␾ x .
Ži. Since H Ž x . F AutŽ X, R . x F , it is clear that r Ž x, x h . s r Ž x k , x h k ..
␾
Since x h s Ž x ␾ . h ␾ s Ž x h . ␾ , it follows from the definition of ␾ that
␾
r Ž x, x h . s r Ž x, x h ..
h
␾
Žii. Since x r Ž x, x . s x h , Ž x h . ␾ 4 s x h , x h 4 , the regularity of H
␾
forces k g h, h 4 .
Žiii. By Ži., r Ž x, x k h . g r Ž x, x h . r Ž x h , x k h . s r Ž x, x h . r Ž x, x k . and
␾
␾
r Ž x, x k h . g r Ž x, x h . r Ž x h , x k h . s r Ž x, x h . r Ž x, x k ..
␾
If r Ž x, x k h . s r Ž x, x k h ., then, by Žii., kh g k ␾ h, Ž k ␾ h. ␾ 4 . Therefore,
k s k ␾ or h s h␾ . This implies that one of r Ž x, x h ., r Ž x, x k .4 is thin and,
hence, < r Ž x, x h . r Ž x, x k .< s 1 by Lemma 2.1Živ.. Since < r Ž x, x h . r Ž x, x k .< F 2
by Lemma 2.1Živ., the conclusion follows.
Lemma 4.3 is frequently used in the remainder of this paper without
further mention.
Let x, y g X and E F O ␪ Ž F .. A group isomorphism ␪ : H Ž E, x . ª
Ž
H E, y . is group-arranged with respect to Ž E, x, y . if ␪ induces an arranged map ␪˜ with respect to Ž E, x, y . by ␪˜Ž x h . [ y ␪ Ž h., h g H Ž E, x .. We
say that E is group-arranged if, for all x, y g X, there exists a grouparranged map with respect to Ž E, x, y ..
The following lemmata are simple notes from group theory.
LEMMA 4.4. Let H be an abelian group of odd order and ␾ g Aut H with
␾ 2 s id. Then H is the direct product of two ␾-in¨ ariant subgroups ² aa␾ N a
g H : and ² ay1 a␾ N a g H :.
Proof. Let a g H. Since a2 s aa␾ Ž ay1 a␾ .y1 and ² a2 : s ² a:, ² a: s
² aa␾ :² ay1 a␾ :. Note that ␾ fixes each element in ² aa␾ : and inverses
each element in ² ay1 a␾ :. This implies that ² aa␾ : and ² ay1 a␾ : are
␾-invariant and ² aa␾ : l ² ay1 a␾ : s id4 . Therefore, we obtain ² a: s
² aa␾ : = ² ay1 a␾ :. Since H is the direct product of cyclic groups, the
conclusion follows.
LEMMA 4.5. Let G be a group and H, K F G. Assume that ␪ : H ª K is
a group isomorphism, a g NG Ž H ., and b g NG Ž K . such that m [
w² a: H : H x s w² b : K : K x. If ␪ Ia s Ib ␪ and ␪ Ž a m . s b m , then ␰ : ² a: H ª
² b : K Ž a i h ¬ b i␪ Ž h., h g H, i g ⺪ G 0 . is a group isomorphism where Ia :
H ª H Ž h ¬ ay1 ha, h g H . and Ib : K ª K Ž k ¬ by1 kb, k g K ..
Proof. The condition of ␪ Ž a m . s b m guarantees that ␰ is well-defined.
It is obvious that ␰ is a bijection. It is routine work to prove that ␰ is a
homomorphism by the condition ␪ Ia s Ib ␪ .
We assume that E F O ␪ Ž F ., x, y g X, and write ␾ [ ␾ x , ␺ [ ␾ y for
the remainder of this section.
QUASI-THIN ASSOCIATION SCHEMES
673
LEMMA 4.6. Assume that ␪ is a group-arranged map with respect to
Ž E, x, y .. If a g H Ž x . normalizes H Ž E, x . and ay1 a␾ g H Ž E, x ., then there
exists a group-arranged map which extends ␪ with respect to Ž E⬘, x, y . where
E⬘ [ C Ž² a: H Ž E, x ...
Proof. Since a␾ g aH Ž E, x ., aH Ž E, x . is a ␾-invariant coset of H Ž x .,
particularly, ² a: H Ž E, x . is ␾-invariant subgroup. Since < aH Ž E, x .< is odd,
there exists b g aH Ž F, x . such that b ␾ s b. Thus, r Ž x, x b . g O␪ Ž F . by the
definition of ␾ . This implies that there exists a unique element c, say, in
H Ž y . such that r Ž x, x b . s r Ž y, y c .. For each i g ⺪ G 0 ,
Ž 3.
!
#
i times
r Ž x, x . 4 s r Ž x, x
b
i
b
½
" !
#
"
i times
. ⭈⭈⭈ r Ž x, x . s r Ž y, y . ⭈⭈⭈ r Ž y, y c .
s r Ž y, y c
i
b
c
.5.
We set m [ w² b : H Ž E, x . : H Ž E, x .x. Since m is the minimal positive
m
integer such that r Ž x, x b . g E, it follows from Ž3. that
m s ² b : H Ž E, x . : H Ž E, x . s ² c : H Ž E, y . : H Ž E, y . .
Furthermore, since b m g H Ž E, x ., r Ž x, x b . s r Ž y, y ␪ Ž b . .. Combining this
m
with r Ž x, x b . g O␪ Ž F ., we obtain from Lemma 4.1Žii. that ␪ Ž b m . s c m .
For each h g H,
m
r Ž x, x h . 4 s r Ž x, x b . r Ž x, x b
b
y1
h
m
y1
. s r Ž x, x b . r Ž x, x h . r Ž x, x b .
s r Ž y, y c . r Ž y, y ␪ Ž h. . r Ž y, y c
y1
. s r Ž y, y ␪ Ž h. . 4 .
c
Since h b g H Ž E, x ., r Ž x, x h . s r Ž y, y ␪ Ž h . .. It follows from Lemma 4.3Žii.
that ␪ Ž h b . g ␪ Ž h. c, ␪ Ž h. c ␺ 4 . We claim that ␪ Ž h b . s ␪ Ž h. c. Assume the
contrary, i.e., that ␪ Ž h b . / ␪ Ž h. c. Then ␪ Ž h b . s ␪ Ž h. c ␺ . We denote by n
the order of b. Since 2 ¦ n and c ␺ s c Žsee the definition of ␺ ., Ž c␺ . n s ␺ .
Therefore,
b
b
␪ Ž h . s ␪ Ž byn hb n . s ␪ Ž h b . s ␪ Ž h .
n
Ž c␺ . n
␺
s ␪ Ž h. .
This implies that ␪ Ž h b . s Ž ␪ Ž h. c . ␺ s Ž cy1 . ␺␪ Ž h. ␺ c ␺ s ␪ Ž h. c, a contradiction.
Since h is arbitrary, we conclude that ␪ Ib s Ic ␪ , where Ib Ž Ic . is the map
on H Ž E, x . Žres. H Ž E, y .. by the conjugation of b Žres. c ..
Note that ² b : H Ž E, x . s ² a: H Ž E, x . since b g aH Ž E, x .. It follows
from Lemma 4.5 that ␰ : ² b : H Ž E, x . ª ² c : H Ž E, y . Ž b i h ¬ c i␪ Ž h.. is a
group isomorphism.
674
MITSUGU HIRASAKA
Finally, we shall prove that ␰ induces an arranged map with respect to
Ž E⬘, x, y .. For each i g ⺪ G 0 and h g H Ž E, x . we have
r Ž x, x
bih
!
#
"
i times
. 4 s r Ž x, x . r Ž x, x . ⭈⭈⭈ r Ž x, x b .
h
s r Ž y, y
½
␪ Ž h.
!
b
#
"
i times
. r Ž y, y . ⭈⭈⭈ r Ž y, y c .
c
5 ½
5
s r Ž y, y c ␪ Ž h. . s r Ž y, y ␰ Ž b h. . .
i
i
This completes the proof since ␰ is a group isomorphism.
PROPOSITION 4.7. Assume that there exists a group-arranged map with
respect to Ž E, x, y .. If a g H Ž x . centralizes H Ž E, x . and ² a: H Ž E, x . is
␾-in¨ ariant, then there exists a group-arranged map with respect to Ž E⬘, x, y .
where E⬘ [ C Ž² a: H Ž E, x ...
Proof. For convenience we set H [ H Ž E, x . and b [ ay1 a␾ . Since
² a: HrH is cyclic, it follows from Lemma 4.4 that
² a: HrH s ² aa␾ : HrH = ² b : HrH.
From the assumption there exist a group-arranged map ␪ , say, with respect
to Ž E, x, y ., and, hence, also I␺ ␪ where I␺ is the conjugation map of ␺
restricted on H Ž E, y ..
We shall prove this proposition by the following steps:
1. We want to find a group isomorphism ␰q or ␰y, say, from ² b : H
²
to c : H Ž E, y . for some c g H Ž y . with r Ž x, x b . s r Ž y, y c ..
1-1. ␪ Ib s Ic ␪ s ␪ .
1-2. Setting m [ w² b : H : H x, we have m s w² c : H Ž E, y . :
H Ž E, y .x and ␪ Ž b m . g c m , cym 4 .
2. ␰q or ␰y induces an arranged map with respect to Ž D, x, y .
where D [ C Ž² b : H ..
3. We want to find a group-arranged map which extends ␰ " with
respect to Ž E⬘, x, y ..
Ž1-1. We may assume that b f H, so that b i f H for each 1 F i F
m y 1. Note that b centralizes H since a, a ␾ g C H Ž x . Ž H ..
Therefore, ␪ Ib s ␪ .
Since b ␾ s by1, r Ž x, x b . s r Ž x, x b .*. Let c g H Ž y . be such that
r Ž x, x b . s r Ž y, y c .. Then, by using induction on i and b ␾ s by1, we obtain
that, for each i g ⺪ G 0 ,
Ž 4.
r Ž x, x b . s r Ž y, y c . .
i
i
QUASI-THIN ASSOCIATION SCHEMES
675
We claim that Ic ␪ s ␪ , i.e., that c centralizes H Ž E, y .. For each h g H,
r Ž y, y ␪ Ž h. . s r Ž x, x h . s r Ž x, x h
b
␾
. g r Ž x, x b . r Ž x, x h . r Ž x, x b . l E
s r Ž y, y c . r Ž y, y ␪ Ž h. . r Ž y, y c
½
s r Ž y, y ␪ Ž h. . , r Ž y, y ␪ Ž h.
c
s r Ž y, y ␪ Ž h.
c
c␺
␺
.5
. lE
.4.
.c
Therefore, ␪ Ž h. g ␪ Ž h , ␪ Ž h. c ␺ 4 . We define
K1 [ h g H N ␪ Ž h. s ␪ Ž h. 4
c
K 2 [ h g H N ␪ Ž h.
c␺
and
s ␪ Ž h. 4 .
Then K 1 and K 2 are subgroups of H such that H s K 1 j K 2 . This
implies that H s K 1 or K 2 , since no group is a union of two proper
subgroups. Therefore, Ic ␪ s ␪ or Ic ␺ ␪ s ␪ . If Ic ␺ ␪ s ␪ , then Ic s I␺ on
H Ž F, y .. Comparing the orders on both sides, we obtain that Ic is the
identity map on H Ž E, y .. Therefore, the claim follows.
m
Ž1-2. Since m is the minimal positive integer such that r Ž x, x b . g E,
m
it follows from Ž4. that m s w² c : H Ž E, y . : H Ž E, y .x. Since r Ž y, y c . s
m
m
r Ž x, x b . s r Ž y, y ␪ Ž b . ., c m g ␪ Ž b m ., ␪ Ž b m . ␺ 4 . According to whether c m
s ␪ Ž b m . or ␪ Ž b m . ␺ we define, respectively,
␰q , ␰y : ² b : H ª ² c : H Ž E, y .
such that ␰q Ž b i h. [ c i␪ Ž h. and ␰y Ž b i h. [ c i I␺ ␪ Ž h. for each i g ⺪ G 0 and
each h g H. Since ␪ Ib s Ic ␪ s ␪ , we conclude from Lemma 4.5 that both
␰q and ␰y are group isomorphisms.
Ž2. For each i g ⺪ G 0 and each h g H,
r Ž x, x b h . g r Ž x, x h . r Ž x, x b
Ž 5.
i
i
.
s r Ž y, y ␪ Ž h. . r Ž y, y c
½
i
.
s r Ž y, y c ␪ Ž h. . , r Ž y, y c
i
yi
␪ Ž h.
.5.
We claim that, if there exists h g H Ž E, x . such that r Ž x, x b h . s
␧
␧
r Ž y, y c ␪ Ž h. . and h ␾ / h where ␧ g 1, y14 , then r Ž x, x b k . s r Ž y, y c ␪ Ž k . .
for each k g H. Assume the contrary, i.e., that there exists a k g H such
␧
y␧
that r Ž x, x b k . / r Ž y, y c ␪ Ž k . .. Then, by Ž5., r Ž x, x b k . s r Ž y, y c ␪ Ž k . ., so
that
r Ž y, y ␪ Ž h k
␾
.
. s r Ž x, x b h b
y1
k␾
. g r Ž x, x b
y1
k␾
. r Ž x, x b h . l E
s r Ž x, x b k . r Ž x, x b h . l E s r Ž y, y c
s Ž y, y ␪ Ž h k . . 4 .
y␧
␪Žk.
. r Ž y, y c
␧
␪ Ž h.
. lE
676
MITSUGU HIRASAKA
Therefore, ␪ Ž hk ␾ . g ␪ Ž hk ., ␪ Ž hk . ␺ 4 . Note that ␪ Ž hk . ␺ s ␪ ŽŽ hk . ␾ . since
␺
␾
r Ž y, y ␪ Ž h k . . s r Ž y, y ␪ Ž h k . . s r Ž x, x h k . s r Ž x, x Ž h k . .. Thus, hk ␾ g
␾
␾
␾
␾
hk, h k 4 and, hence, k s k or h s h . By the assumption, k s k ␾ and,
hence,
< r Ž x, x k . r Ž x, x b . < s < r Ž y, y ␪ Ž k . . r Ž y, y c . < s 1,
␧
contradicting r Ž x, x b k . / r Ž y, y c ␪ Ž k . ..
i
␧i
Second, we claim that r Ž x, x b k . s r Ž y, y c ␪ Ž k . . for each k g H and
each i with 0 F i F m y 1 under the same assumption as in the above
claim. Use induction on i. The above claim and the property of ␪
guarantee the validity when i F 1. Assume that i is the minimal number
i
␧i
such that r Ž x, x b k . / r Ž y, y c ␪ Ž k . . for some k g H and 1 - i F m y 1.
Since
r Ž x, x b k . g r Ž x, x b
i
½
iy 1
s r Ž y, y c
␧i
k
. r Ž x, x b . s r Ž y, y c
␪Žk.
. , r Ž y, y c
␧ Ž iy 2.
␪Žk.
␧ Ž iy 1.
␪Žk.
.5.
. r Ž y, y c .
it follows from the choice and minimality of i that
r Ž x, x b k . s r Ž y, y c
i
␧ Ž iy 2.
␪Žk.
. s r Ž x, x b
iy 2
k
..
Therefore, b i k g b iy2 k, Ž b iy2 k . ␾ 4 and, hence, b 2 s 1 or b 2Ž iy1. s k ␾ ky1.
From the definition of b and 2 ¦ m, b iy1 g H, contradicting that 1 - i F
m y 1.
We shall prove that ␰q or ␰y induces an arranged map with respect to
Ž E, x, y ..
If each element in H is fixed by ␾ , then, for each b i h g ² b : H,
r Ž x, x b h . 4 s r Ž x, x h . r Ž x, x b . s r Ž y, y ␪ Ž h. . r Ž y, y c . s ½ r Ž y, y c ␪ Ž h. . 5 .
i
i
i
i
Therefore, since ␰q is a group isomorphism, ␰q induces a group-arranged
map with respect to Ž D, x, y ..
If there exists a nonfixed point h, say, in H by ␾ , then, by Ž5.,
½
r Ž x, x b h . g r Ž y, y c␪ Ž h. . , r Ž y, y c
y1
␪ Ž h.
5.
y1
Choose ␰q if r Ž x, x b h . s r Ž y, y c␪ Ž h. . or ␰y if r Ž x, x b h . s r Ž y, y c ␪ Ž h. .. By
the second claim in the argument Ž2., ␰q or ␰y, induces a group-arranged
map with respect to Ž D, x, y ..
Ž3. Thus, there exists a group-arranged map with respect to Ž D, x, y ..
Since aa␾ normalizes H Ž D, x . and Ž aa␾ .Ž aa␾ .y1 g H F H Ž D, x ., it follows from Lemma 4.6 that there exists a group-arranged map with respect
to Ž C Ž E⬘., x, y .. This completes the proof.
QUASI-THIN ASSOCIATION SCHEMES
COROLLARY 4.8.
677
If H Ž x . is abelian, then O ␪ Ž F . is group-arranged.
Proof. For convenience we set
H [ H Ž x . , ␾ [ ␾ x , L [ ² aa␾ N a g H : , M [ ² ay1 a␾ N a g H : .
By Lemma 4.4, H s L = M. Note that each element of L is fixed and
each element of M undergoes inversion by the conjugation of ␾ . Applying
Lemma 4.6 and Proposition 4.7 for cyclic groups in L and M, the
conclusion follows from the fact that both L and M are the direct
products of cyclic groups.
THEOREM 4.9.
arranged.
If F g C Ž R . is quasi-thin and O␪ Ž F . s 14 , then F is
Proof. By Lemma 4.2Žiii., O ␪ Ž F . s F. Therefore, by Corollary 4.8, it is
sufficient to show that H Ž x . is abelian.
By Lemma 4.2Žii., F is symmetric. Therefore, h␾ x s hy1 for each
h g H Ž x .. Let h, k g H Ž x . be two arbitrary elements. Since F is commutative, r Ž x, x h . r Ž x, x k . s r Ž x, x k . r Ž x, x h .. It follows from Lemma 4.1 that
r Ž x, x k h . , r Ž x, x k
y1
h
. 4 s r Ž x, x h k . , r Ž x, x h
y1
k
.4
and, hence, kh g hk, Ž hk .y1 , hy1 k, ky1 h4 . Since each element of H Ž x .
has odd order, the condition kh g Ž hk .y1 , hy1 k, ky1 h4 implies that one
of hk, h, k is the identity. Therefore, kh s hk. This completes the proof.
5. APPLICATIONS
THEOREM 5.1. Let Ž X, R . be a quasi-thin scheme and G be gi¨ en in
Proposition 2.2. If G is meta-cyclic and 4 ¦ < G <, then O ␪ Ž R . is grouparranged. Furthermore, AutŽ X, R . is transiti¨ e on X.
Proof. If R is thin, then O ␪ Ž R . s 14 and AutŽ X, R . is regular on X
Žsee w9, p. 177x..
We assume that R is not thin. Combining Proposition 2.2Ži. with the
assumption 4 ¦ < G <, we conclude that the order of G is twice an odd
number. By w8, Theorem 4.6x, G has a unique subgroup H, say, of index
two. Since G is meta-cyclic, there exists a, b g G such that ² a: 1
᎐ G and
G s ² a:² b :.
Note that G has at least two involutions Že.g., ␾ x / ␾ y if r Ž x, y . f
O␪ Ž R .. and ² a: has at most one involution. It follows from Sylow’s
Theorem that ² a: has no involution. This forces w G : ² a:² b 2 :x s 2. Therefore, we conclude from the uniqueness of H that H s ² a:² c : where
c [ b2.
678
MITSUGU HIRASAKA
Let x, y g X be two arbitrary points. We set ␾ x [ ␾ and d [ cy1 c ␾
for short. Since ² a: 1
᎐ G, ² a: is ␾-invariant by conjugation. Applying
Proposition 4.7 for a g CH Žid G ., we obtain that there exists a grouparranged map with respect to Ž C Ž² a:., x, y .. It is well known that Aut² a: is
abelian. Since Id s I␾ d ␾ s Id y1 , Id 2 is the identity map on ² a: where
Id Ž I␾ d ␾ . are the conjugation maps of d Žrespectively, ␾ d ␾ . restricted on
² a:. This implies that d centralizes ² a:. Applying Proposition 4.7 for
d g CH Ž² a:., we obtain that there exists a group-arranged map with
respect to Ž E, x, y . where E [ C Ž² d :² a:..
Since Hr² a: is cyclic, it follows from Lemma 4.4 that Hr² a: s
² cc ␾ :² a:r² a: = ² d :² a:r² a:. Since
Ž cc ␾ .
y1
␾
␾
Ž cc ␾ . s Ž cy1 . cy1 c ␾ c g ² d :² a: ,
it follows from Lemma 4.6 that there exists a group-arranged map with
respect to ŽO ␪ Ž H ., x, y .. Therefore, O ␪ Ž R . is group-arranged; particularly,
O ␪ Ž R . is arranged.
␪
We claim that R s O ␪ Ž R .O␪ Ž R .. Since Ž X, R . O Ž R. is thin Žsee w9,
Theorem 2.3.1x., we conclude from Ž1. that, for each r g R,
n O ␪ Ž R. r O ␪ Ž R. s n O ␪ Ž R. r s n O ␪ Ž R. .
Since 2 ¦ n O ␪ Ž R. and R is quasi-thin, there exists t g O ␪ Ž R . r l O␪ Ž R .,
implying that r g O ␪ Ž R . t : O ␪ Ž R .O␪ Ž R ..
Thus, it follows from Lemma 3.2 that R is arranged. By Lemma 3.1,
AutŽ X, R . is transitive on X.
COROLLARY 5.2. Let Ž X, R . be a quasi-thin scheme such that 2 ¦ < X <
and n O ␪ Ž R. is square-free. Then AutŽ X, R . has a regular automorphism group.
Proof. If R is thin, then it is well-known that AutŽ X, R . is regular on X
Žsee e.g. w9, p. 177x..
We assume that R is not thin, so that there exists r g R such that
sr / 1. If sr g O␪ Ž R ., then, by Lemma 4.1, ² sr : s 1, sr 4 and, hence,
n ² s r : s 2. From Ž2. we obtain 2 N < X <, a contradiction. Therefore, n s r s 2,
particularly, O ␪ Ž R . ­ O␪ Ž R ..
Applying Proposition 2.2Žii, iii., we obtain that < G < s < Gx < < x G < s 2 n O ␪ Ž R. ,
where G is given in Proposition 2.2. From Ž2. we obtain that 2 ¦ n O ␪ Ž R. ,
implying 4 ¦ < G <. It is known that any group of square-free order is
meta-cyclic Žsee w3, Section 9.4x.. Applying Theorem 5.1 to Ž X, R ., we
conclude that AutŽ X, R . is transitive on < X <. Let x g X be a point. Since
ŽAutŽ X, R .. x s ² ␾ x : by Proposition 2.2Žiii., <AutŽ X, R .< s 2 < X <. By w8,
Theorem 4.6x, AutŽ X, R . has a unique subgroup H, say, of index two. Since
< H < s < H x < < x H < s < X < and 2 ¦ < X <, < H x < s 1 and, hence, < x H < s < X <. This
implies that H is regular on X.
QUASI-THIN ASSOCIATION SCHEMES
679
ACKNOWLEDGMENT
The author expresses his deepest gratitude to Professor Bannai, Professor Munemasa,
Professor Muzychuk, and an anonymous referee for their advice and encouragement. This
paper owes much to their thoughtful and valuable comments.
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