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On R-Sequenceability and Rh-Sequenceability of Groups
Annals of Discrete Mathematics 18 (1983) 535-548 0 North-Holland Publishing Company
535
ON R-SE()UENCEABILITY AND Rh-SEQUENCEABILITY OF GROUPS A.D.
Keedwell
ABSTRACT A f i n i t e group (G,.)
o f o r d e r n i s s a i d t o be R-sequenceable ( o r near-se-
quenceable) i f i t s elements
a o y aly
..., a n-1
t h e p a r t i a l p r o d u c t s bo = a o y bl = aoaly
can be o r d e r e d i n such a way t h a t
...,bn-2
b2 = aoala 2y...y
a r e a l l d i f f e r e n t and so t h a t t h e p r o d u c t bn-l
=
aoala2...an-l
= a a a
= b
0
o 1 2**'an-2 We show
= e.
t h a t a f i n i t e d i h e d r a l group i s R-sequenceable i f and o n l y i f i t i s o f d o u b l y even o r d e r and t h a t a n o n - a b e l i a n group o f o r d e r pq such t h a t p has 2 as a p r i m i t i v e r o o t , where p and q a r e d i s t i n c t odd primes w i t h p
< q, i s R-sequenceable.
We also
d i s c u s s t h e g e n e r a l i z e d concept o f R h - s e q u e n c e a b i l i t y and i t s r e l e v a n c e t o t h e c o n s t r u c t i o n o f s e t s o f m u t u a l l y o r t h o g o n a l l a t i n squares.
1. INTRODUCTION We b e g i n w i t h some d e f i n i t i o n s .
A f i n i t e group (G,.) o f o r d e r n i s s a i d t o be sequenceable i f i t s elements e, aly
can be a r r a n g e d i n a sequence a.
a2,
..., a n-1
i n such a way t h a t t h e
are a l l = a a a n-1 o 1 2**'an-1 d i s t i n c t (and consequently a r e t h e elements o f G i n a new o r d e r ) . I t i s s a i d t o p a r t i a l p r o d u c t s bo = a o y bl = aoaly
b 2 = aoala 2y...yb
be R-sequenceable (see [ 21) o r near-sequenceabk
(see [ 51) i f i t s elements can be
o r d e r e d i n such a way t h a t t h e p a r t i a l p r o d u c t s bo = a o y bl = aoaly b
2 bn-l
a r e a l l d i f f e r e n t and so t h a t t h e p r o d u c t o 1 2.~e~n-2 = a a a = b =e. o o 1 2***an-1 A one-to-one mapping g + O ( g ) o f a f i n i t e group (G,.) o n t o i t s e l f i s s a i d t o
= aoa,a 2y...yb
n-2
= a a a
be a compZete mapping i f t h e mapping g
+
$ ( g ) , where $ ( g ) = g . o ( g ) i s a g a i n a one-
I f (Gym) i s R-sequenceable, t h e n t h e mapping
to-one mapping o f G o n t o i t s e l f . -1 f o r i = 1,2,...,n-2,e(bn-l) = al&(c) = a o ( b i ) = b i. b i+l i+l t h e element o f G which does n o t o c c u r i n t h e s e t {bo,bl,b2,...,b
= a
0
n-1
= e, where c i s
1, i s a com-
536
A.D. Keedwell
p l e t e mapping, as was shown by
L.J. Paige i n [ 7 ] .
Thus t h e c o n d i t i o n t h a t (G,.)
be R-sequenceable i s a s u f f i c i e n t c o n d i t i o n f o r G t o have a complete mapping b u t i t i s n o t always necessary. btl-l
For example, f o r a b e l i a n groups, t h e c o n d i t i o n
= e i s alone s u f f i c i e n t (L.J.
Paige [6]). When and o n l y when a group G has a
complete mapping, t h e l a t i n square formed by the Cayley t a b l e o f G has an orthogonal mate (see s e c t i o n 1.4 o f [ 1 ] f o r a p r o o f ) and, i f t h e group i s R-sequenceable, t h i s orthogonal mate can be constructed by t h e procedure c a l l e d t h e c o l m r*:=;rm> i n [ 4 ] .
(See a l s o s e c t i o n 7.4 o f [ 11).
The column method makes use o f the
permutation 6 d e f i n e d above which we can w r i t e i n c y c l e form as ( c ) ( b l b2 b3 -1 -1 -1 -1 bn-l) and f o r which t h e elements bl b2, b2 b3, bn-2bn-1, bn-lbl a r e the
...,
-1 -1 I f f u r t h e r bl b3, b2 b4,
...
-1 ..., bn-2bl,
b - l b i s again n-1 2 a r e - o r d e r i n g o f t h e n o n - i d e n t i t y elements o f G, then t h e column method permits n o n - i d e n t i t y elements o f G.
t h e c o n s t r u c t i o n o f a t l e a s t t h r e e m u t u a l l y orthogonal l a t i n squares based on the Cayley t a b l e o f G. More g e n e r a l l y , we s h a l l say t h a t a group (G;) a!Zt
o f order n i s Rh-scquenca-
i f n-1 o f i t s elements can be arranged i n a sequence cl,
c2,
..., c n-1
in
such a way t h a t the s e t o f elements c-’ciil f o r i = 1,2,...,n-l, are a l l d i s t i n c t i (where a r i t h m e t i c o f s u f f i c e s i s modulo n-1), and l i k e w i s e the s e t s o f elements -1 -1 -1 c . c i + 2 ’ c . Ci+3’ c 1. c i + h ’ I n p a r t i c u l a r , a group which i s R 1-sequenceable 1 i s R-sequenceable w i t h -1 -1 -1 -1 = c c a = e and al = c ~ - ~ ca2~ =, c c 2 , a3 = c2 c3, a 0 n-1 n-2 n-1’ I f a group ( G , . ) i s Rh-sequenceable, the column method permits t h e c o n s t r u c t i o n o f
...,
,
..., ...@
a t least h
+ 1
m u t u a l l y orthogonal l a t i n squares based on t h e Cayley t a b l e o f G
( a s has been shown i n [ 41 and [ 1 1 ) . R. Friedlander, 8. Gordon and M.D. t l i l l e r have i n v e s t i g a t e d the R-sequencea b i l i t y o f various classes o f a b e l i a n groups i n [ 2 ] .
Here we o b t a i n some r e s u l t s
on the R-sequenceability o f c e r t a i n non-abelian groups and make a few observations about Rh-sequenceability. I n t h e i n v e s t i g a t i o n o f s e q u e n c e a b i l i t y o f groups, t h e concept o f a quotient ne~7~lenoing has played an i m p o r t a n t r o l e .
We s h a l l make use o f t h e same concept i n
t h e present paper.
DEFINITION: I f
a i s any sequence u
, u2,
..., un o f elements o f a group G,
P(.x) w i l l denote i t s sequence o f p a r t i a l products v1 = ul,
v2 = u1u2,
then
531
On R-sequenceability and Rh-sequenceability of groups v
= u u u3 a . .
.,..., vn = u1u2.. .u n'
L e t G be a non-abelian group o f order n
which has a normal subgroup H o f order w, and l e t G/H = gp I X ~ , Xwhere ~,...,X~} t = n/w.
A sequence a o f length n c o n s i s t i n g o f elements o f G/H i s c a l l e d a
quotient sequencing o f G i f each x
i' The image under the mapping $:G
1
Q
-+
i Q t, occurs w times i n both a and P(a).
G/H o f e i t h e r a sequencing o r a near-se-
quencing o f G i s a q u o t i e n t sequencing o f G.
I n the l a t t e r case, the l a s t e n t r y
i n t h e q u o t i e n t sequencing i s necessarily the i d e n t i t y element o f G/H.
2. R-SEQUENCEABILITY OF DIHEDRAL GROUPS n 2 The dihedral group On = gpCa,8 : a = E , B = E , aB =
Ba
-1
1 o f order 2n cannot
be R-sequenceable i f n i s odd since the product o f a l l i t s elements takes the form n s -1 B CY f o r some i n t e g e r s , 0 Q s < n, when the r e l a t i o n a6 = @a i s used t o s i m p l i f y i t and t h i s cannot be equal t o the i d e n t i t y element unless n i s an even integer.
We have been able t o show t h a t t h i s necessary c o n d i t i o n f o r R-sequence-
a b i l i t y i s also sufficient.
That i s , the dihedral group Dn o f order 2n i s R-se-
quenceable i f and only i f n i s an even integer. quencing i n FIG. 1.
We e x h i b i t a s u i t a b l e R-se-
This i s based on the q u o t i e n t sequencing
1 1 . . . 1 m
x
1 1 . . . 1
1's
m
1's
x
x . . . x
(2m-1)
x's
n 2 where n = 2m, H = gpCa : a = € l a n d D /H = { l a x ] . The cases n = 4h and n = 4h n have t o be t r e a t e d separately. (The R-sequenceability o f Dn f o r the p a r t i c u l a r
-
values n = 2,4,6,8,10,14,20,22,32,34,36,38,50,64,66
and f o r eighteen other values
between n = 100 and n = 4084 has already been established i n [ 211 .)
3.
R-SEQUENCEABILITY OF NON-ABELIAN GROUPS OF ORDER pq
I n a recent paper [ 5 1 , we showed t h a t the non-abelian group o f order pq, where p and q are d i s t i n c t primes w i t h p prime which has 2 as a p r i m i t i v e element. also R-sequenceable.
< q, i s sequenceable whenever p i s a We show next t h a t these same groups are
Our method i s almost a copy o f t h a t described i n [ 51 and
uses the same q u o t i e n t sequencing. L e t H be the unique normal subgroup o f order q i n the given group G o f order pq and suppose t h a t the f a c t o r group G/H i s generated by the element x.
A neces-
A.D. KeedweN
538
+
sary c o n d i t i o n f o r t h e existence o f G i s t h a t q = 1 i n t e g e r h.
2ph f o r some p o s i t i v e
We suppose t h a t p has 2 as a p r i m i t i v e element and l e t u s a t i s f y t h e
congruence 20. h oh 5 1 * 2
= 1 mod p, f
1 mod p.
Then o i s a p r i m i t i v e element o f GF[p] s i n c e The f o l l o w i n g i s a q u o t i e n t sequencing f o r G (as i s
shown i n [ 51):
a sequence o f 2ph l ' s , f o l l o w e d by x, f o l l o w e d by a sequence o f 2ph-1 p-2 p-3 p-2 2 3 2 0-1 5 -u 0 - 0 0 -u 1-0 (p-1)-tuples x x X x X , where i n d i c e s a r e computed
...
modulo p, f o l l o w e d by a sequence 0-1 xu2-a 03-02 ap-2-0p-3 X X x
...
1 x
2
x
4
x
8
... x 2P-2
l-oP-2 X
2 xu Jp-3
*
The p a r t i a l products a r e 2ph l ' s , f o l l o w e d by the sequence x x' 2 p-2 p-2 repeated 2ph-1 times, and then the sequence x x' xu XU x' 2 x' xu x 1. (We use the f a c t t h a t o ~ = - 2.)~
...
X
2-2
... x
p-4
X0
...
L e t G be generated by two elements a, b such t h a t a' ab = ba
S
, where
sp
9
= e, bp = e and
1 mod q and e i s t h e i d e n t i t y element o f G.
Then,
X
(b'av)(bxaY)
u+x vs +y a and G has a unique normal subgroup H = gpIa : aq = e l o f
= b
The n a t u r a l homomorphism G 2 elements 1 = H, x = bH, x2 = b H,
order q.
+
G/H maps G o n t o t h e c y c l i c group C
..., xP-'
= bP-lH.
P
with
L e t r be a p r i m i t i v e element o f GF[q]. Then (compare [ 5 1 ) we seek a nearh-2 ,h-3 h h-1 ,h+l-,h r ,a , a 9 by sequencing o f G i n t h e form e, ar
-'
11
= r
...,
h-1- h-2 r . We make use o f t h e f a c t s t h a t oP-r
t h a t ap-r-Gp-r-l
= -oP-r
, where
2a
= Zr-',
t h a t l-oP-'
= -1,
and, generally,
1 mod p.
This sequencing g i v e s r i s e t o t h e p a r t i a l products which a r e l i s t e d i n vw FIG. 2 . (For ease o f p r i n t i n g , we have represented t h e element b a by the The expressions denoted by E ( j ) ( f o r i and j = O,l,...,p-2) i n FIG. 2 a r e as given below. We equate
ordered p a i r v, w i n FIG. 2.)
i = 1,2,...,q-2
these expressions t o expressions i n a p r i m i t i v e element t o f G R q ] e x a c t l y as i n
I 5 I.
We a l s o make t h e d e f i n i t i o n s a!?! = t a ( J ) f o r i = l,Z, ...,qi
2 and
5 39
On Rsequenceability and Rh-sequenceability of groups
... +...+a
+a
it1 U
= t
p-2
it1
Q 1-1 i-1 ( t -t )t
.
From these e q u a l i t i e s , we o b t a i n E(l)-E(o)su-l 1 1
U
-a2
= ( t l-su-l)(ta-ta-l),
1
2
u u E(2)-E(1)su -a= (2)-a(2) = ( t 2-t '5 1 1 a2 1
2
-a)(te-tL-l),
... E (P-2) -E(P-3 1 1
p-2
Iso-a
tE(o)-E(P-2)s1-o 1 1
P-2
p-3
(P-2)-a(P-2) = a2 1
-
t a ( P - ' )-a(P-l) 1 1 U
Hence, a") 1
= t"-'(t
U
= ( t P-2-t up-3 ap-2-up-3 U
= ( t - t p-2S1-ap-2
Q a-1 ) ( t -t
L 11-1 ) ( t -t ) Y
1.
1 0-1 -s ),
( 2 ) = t a - l ( t u2 -t o2-0 al s ), .(3) 1
u u 3 2 ( t 3-t 2s u -a 1,
=
...
(P.2) al
=
U
P-2-t
u P-2-aP-3 P-3Sa
) Y
U
(P-1) = ta-l (t-t al
I t i s easy t o check, using FIG. 2 , that, w i t h these choices f o r the expres-
sions E(k) and the indices a(') i i
i n terms o f the p r i m i t i v e element t o f GF[q], a l l
5 40
A.D. Keedwell
t h e p a r t i a l products o f the form b'
j
(j) aB1 a r e d i f f e r e n t i n c l u d i n g t h e l a s t one
U
which occurs provided t h a t
p =
( T h i s has been i l l u s t r a t e d i n [ 51).
t p-2(-te-2)).
F i n a l l y , i t i s easy t o see t h a t a l l b u t the l a s t o f t h e p a r t i a l products We r e q u i r e i n a d d i t i o n t h a t t h e l a s t
which a r e powers o f a alone a r e d i f f e r e n t .
such product i s equal t o t h e i d e n t i t y element e. -I:
t
-2
p l ) ( l - s - l ) + LIS 9- 1
That i s we r e q u i r e U
0 mod q.
u
Using t h e f a c t t h a t
= t p - 2 ( - t e - 2 ) and
U
that
= t'-'(t-t
9- 1
'-'s-'),
U
t h i s becomes
U
t P-2te-2ttL-2(t-t p-2s-1)(i-s-1)
Since u = r
h-1
-r
-
t
Up-2 1-2
t
s
-2-
- 0, whence t U p-2-1 = -1. U
h-2 . h-1 ,h-2 i n FIG. 2, we must a l s o have r
-
= t p-2(-te-2).
That
We conclude t h a t an R-sequencing (near-sequencing) o f t h e non-abelian group o f order pq, where 2 i s a p r i m i t i v e element o f GF[pl, can be obtained by t h e f o l l o w i n g procedure. F i r s t choose one o f t h e r o o t s s o f t h e congruence sp compute the i n t e g e r o such t h a t 20
=
1 mod q and a l s o
Then choose p r i m i t i v e elements r up-2-1 such t h a t , modulo q, t = -1 and and t of GFlqI and i n t e g e r s u1,u2,...,u U u -u 2 u -u y 2 2 U aP-2-uP-3 1 so-l 2 1 3 2 su - u p 2-'p -3 s a -u , t t # s ( s i n c e we
.
1 mod p.
,..., -
~
r e q u i r e a ( J ) # 0 f o r j = 1,2, ...,p- 1 ) . Compute the i n d i c e s a ( J ) f o r i 1 9,-1 i = 1,2 q-1 and j = 1,2 p-1. F i n a l l y choose h so t h a t r h - 2 ( r - l ) = t
,...,
modulo q .
,...,
This gives an R-sequencing f o r a r b i t r a r y choice o f t h e index 9,.
I n order t o i l l u s t r a t e t h e procedure, we have c a r r i e d o u t t h e necessary c a l c u l a t i o n s f o r the case p = 5 and q = 11 and we d i s p l a y t h e r e s u l t i n g nearsequencing o f t h e non-abelian group o f order 55 i n FIG. 3. Since 20
1 mod p, we have u = 3.
We may choose s = 3 ( s i n c e 35
and r = t = 2 ( s i n c e 2 i s a p r i m i t i v e element o f GFI111).
t5
-1 mod 11 so u
3
= 6.
u u zero ifwe choose t ' = t
We f i n d t h a t a \ ' ) , = 1.
We a l s o p u t t
a{3'
!L-1
= 1.
1 mod 11)
Then (and
al( 4 ) )
are a l l non-
Then u = 1 = 2
h-2
(2-1),
so
541
On R-sequenceability and Rh-sequenceabifity of groups U
h = 2.
We o b t a i n
1 0-1 = 1 -s
a:’)
= t
.(2) 1
= t 2-t
u
(4) = t .l
-
u
2 $0
-
3
-u= 1
2
-
= 3, 36 = 9 ,
u 3 t 3 s 1-0 = t ( l + s -1 ) = 2(1+4) = -1,
where these l a t t e r c a l c u l a t i o n s a r e made modulo 11. We a l s o have t h e group 2 2 9 4 4 4 g e n e r a t i n g r e l a t i o n s ab = ba3, ab = b a , ab3 = b3a5, ab = b a
.
4. R,-SEQUENCEABILITY I n g e n e r a l , t h e problem of d e t e r m i n i n g which groups a r e Rh-sequenceable seems t o be a d i f f i c u l t one.
F o r a b e l i a n groups G, i t i s c e r t a i n l y necessary
t h a t t h e Sylow 2-subgroup o f G s h o u l d be e i t h e r t r i v i a l o r n o n - c y c l i c . However, even f o r c y c l i c groups, t h e problem i s n o t t r i v i a l . group i s R-sequenceable i f and o n l y i f i t has odd o r d e r . G. R i n g e l and a l s o has been proved i n 121. C4mtl
2m-3,
Certainly a cyclic
T h i s has been s t a t e d by
The f o l l o w i n g R-sequencings for
and C4mt3
are a l t e r n a t i v e t o those given i n t h e l a t t e r reference.
F o r C4m+l
an R-sequencing i s
...,..., 2m-2,
0 4m, 2, 4m-2, 4, 4m-4, 6, 4m-6, 2mt5,
...,..., 2m-(2m-3),
F o r C4m+3
2m+(2m-1),
(See [ 3 ] . )
4m-(2m-2),
2m-(Zm-l),
2m, 2m-1, 2mt3,
2m+l.
an R-sequencing i s
0, 4mt2, 2, 4m, 4, 4m-2, 6, 4m-4, 2m-1, 2m+5, 2m-3,
...,..., 2m-2,
...,..., 2m-(2m-3),
4m-(2m-4),
2mt(2m+l), 2m-(2m-l),
Zm, 2m+l, 2m+3, 2mt2.
However, a computer search has shown t h a t C9 i s n o t R2-sequenceable,
A
s i m i l a r search has shown t h a t C15 i s n o t R3-sequenceable b u t t h a t i t has 32 i s o m o r p h i c a l l y d i s t i n c t R -sequencings. The c y c l i c group CPl i s R2-sequenceable b u t 2 a t t h e p r e s e n t t i m e t h e q u e s t i o n whether i t i s a l s o R3-sequenceable remains undecided. The R h - s e q u e n c e a b i l i t y o f c y c l i c groups o f p r i m e o r d e r i s covered by t h e f o l l o w i n g theorem: n THEOREM: An eZernentary abeZian group of order p i s Rh-sequemeable for
5 42
A.D. Keedwell
...,p n -2.
h = 1, 2,
i s a p r i m i t i v e element o f t h e G a l o i s F i e l d GF[pn]
PROOF: I f
, the
f o l l o w i n g se-
quencing o f t h e elements o f t h e a d d i t i v e group GF[pn] ( w h i c h i s e l e m e n t a r y a b e l i a n ) has t h e r e q u i r e d p r o p e r t i e s :
0 , w-1,
UJ
2
3
2
u -w
-w,
,
...,
w
n p -2
(The n-1 elements c a r p t h e n 0, ~
1
n
p -3
-0
, 1-w pn-2 .
given i n t h e d e f i n i t i o n o f Rh-sequenceability n 2 o r 1, UI, w , , up -2.)
...
,
T h i s g i v e s a t once t h e well-known r e s u l t t h a t n
n
COROLLARY: r'rotr an e?smentary abelian group of order p a s e t o f p -1 w t u u l l y staz2ogonal lati.? squarss all Lased on t h a t group em be constructed and hence a n
;csai2juaotan pv,it.cti20 plans o f order p ,
EXAMPLE: R -szqusnceatiLLty of C 7 3
c3'
2 8 I n GFI 91, t h e r e e x i s t s a p r i m i t i v e element w such t h a t o = w + 1, w = 1. 3 We can w r i t e C3 x C 3 = { a : a3 = e l x ;b : b = e l o r , i n a d d i t i v e n o t a t i o n , as
i
io,1,-1
! + I
I.
{O,b,VJ
Our R -sequencing i s
7
0, w-1,
w
2
-UJ,
o r 0, 0-1,
1,
3 lL
2
-w
,
O, w t l ,
4
I)
-w
3
, w 5-w 4 , w 6 -w 5 , w 7 -w6
- ~ + l ,-1,
-w,
y
1-w
7
- ~ l .
I n m u l t i p l i c a t i v e n o t a t i o n , we have t h e sequencing 2 2 2 2 2 e , a b, a, b, ab, ab2, a2, b , a b w i t h p a r t i a l p r o d u c t s e, a2b, b y b2, a, a b , 2
n
ab
L
, ab,
e.
Thus t h e sequence o f n-1 elements c
1 ' c2; d e f i n i t i o n o f R h - s e q u e n c e a b i l i t y becomes e, a2b, b, b group.
..., c n-1
, a,
given i n the
a2b2, ab2, ab f o r t h i s
The f i r s t " d i f f e r e n c e s " c f ' ci+l a r e as l i s t e d i n t h e sequencing above. -1 2 2 2 2 The second " d i f f e r e n c e s " c . a r e b, ab, ab , a2, b2, a b , a b, a; and so on. 1
5 43
On R-sequenceability and Rh-sequenceability of groups 4h-2
e a
2h-1
e
e a
2h-1
- ( 2h-4)
a
a
a
a a
2e-1 a -(2a-2) a
a
a
-2 a
a B
a
2h-2
h+k-1
a
- ( 211-3) a
2 a
h
a
-1 a
3h-1
a
3h+l
a
3h-2
a
Ba
6a
-2 Ba Ba
Ba
-(2e-2)
3h-2 3h-1 3h-3 3h
3h+t-3 3h-9.-1
Ba
a
22-1 Ba
3htt-2
2h
-(2h-1) -1
Ba
3
3h+.t-1
Ba
h-1
Ba
3h-a+l
3h-n
h-k+l
4h-3
Ba
Bcr
h+e-2
h a
3h
2a-2
2
h- 2
h-1
f3a
2h-2
21-2 a
h+l
Ba
2h-3
a
h-k+l
Ba
a
a
h-.t
h-E
2
-(2a-1)
2h-2
a
2h-4 a
a
Ba
a
a
-(2h-5)
Ba
-3
a
2
-1
a
e 2h-2
a
a
a
I
- ( 2h-3)
-(2h-2) 2h-3
P a r t i a l products
Ba
2h-3
a
FIG. 1 (continued over.)
4h-3
Ba
A .D.Keedwelt
544 R-sequencing o f D 4h 0
J
P a r t i a l products
-(2h-1) 6a
-2h
2h
B.1
2h
2h 6a
2h+l 6a
2ht2
ii
ea
4h-2
3
2h+l
6a
6a
h
Ba
5a
a
2 5a 3
a.1
3h+l
4h-5
e
3h-3 h-1
a
h
Ba
2h- 1
2h
3h-1
B
6a
2
3h
4h-3
4h-1
5a
Ba
%a
3h+2
2h-1 6a
2h-1
a
4h-4
a
a
2h-2 6s
5
a
6a
a
2h-1 5a
a
h+l
5a
P a r t i a l products
h-2 6a
3h-1
4h-1
6s
i3a
2h+l
h- 1
a
BCI
2h- 1
2h-1 6a
a
En
- (2h-2)
a a
6
2h
?-sequencing o f D 4h-2
2h-3
5a
2h-2 %a
FIG. 1 (continued)
2h-2
%a
e
On R-sequenceability and Rh-sequenceability of groups
0, 0
h
0, r -r 0, r
h-1
h t l rh
-
0, r h-2 - ,h-l
-usa
!J-2
_= -
u
(l)sap-*-a
+a1
(24u +al
p-2
-a
2 t. * .+a
(P-34a 1
p-2
FIG. 2 (contd. on next p a g e )
-0
p-3
(P-2)
+a 1
5 45
5 46
A.D. Keedwell
+.
p-2 p-3 (P-2)+qc2 ..+a 1(P-3)su -' +al i=l
f . . .+a (P-3)s0 1
p-2
p-3
E(Pe2) i
+,(p-')+qi2 E(P-2)+,, 1 i=li P-2
+...+a
1
+...+a
1 P-3 +a
FIG. 2 ( c o n t d )
On Rsequenceability and Rh-sequenceabiHty of groups
Sequencing
7 e 2
a a a a
4
8
5
a
a a a a
10 9 7 3 6
b 2 3 b a 9 ba 3 4 b a 4 10 b a 2 6 b a 7 ba 3 8 b a 4 9 b a 2 b a 3 ba 3 5 b a 4 7 b a 2 2 b a 6 ba 3 10 b a 4 3 b a 2 4 b a ba 3 9 b a 4 6 b a 2 a b a 2 ba
e 2
a
6
a
3
a a
8
a a
7 5
a a a
4 10
baa 3 9 b a 4 3 b a 2 a b a 9 ba 3 10 b a 4 4 b a 2 6 b a b 3 b a 4 6 b a 2 2 b a 4 ba 3 5 b a 4 10 b a 2 5 b a ba 3 2 b a 4 7 b a b2 6 ba 3 7 b a 4 b a
Sequencing
P a r t i a l products
(contd) 3 7 b a 4 b a 2 5 b a 4 ba 3 3 b a 4 2 b a 2 10 b a 8 ba 3 6 b a 4 4 b a 2 9 b a 5 ba 3 b a
(contd)
2 b a 5 ba 3 6 b a b4 2 3 b a 3 ba 3 4 b a 4 9 b a 2 7 b a
ba
10
b3 4 5 b a 2 4 b a
4 8
2 ba 3 3 b a 4 8 b a 2 9 b a 2 10 b a 4 2 b a 3 8 b a 7 ba
b a 2 7 b a 10 ba 3 2 b a a
b2 b4 b3 4 5 b a
e
FIG. 3
5 41
A.D. Keedwell
548
BIBLIOGRAPHY 1. 2.
3.
4. 5.
6. 7.
J. Denes and A.D. Keedwell, "Latin Squares and their Applications". (Akademiai Kiad6, Budapest/English Universities Press, London/ Academic Press, New York, 1974.) R.J. Friedlander, 6. Gordon and M.D. Miller, On a group sequencing problem of Ringel. Proc. Ninth S.E. Conf. on Combinatorics, Graph Theory and Computing. Florida Atlantic Univ., Boca Raton, 1978. (Congressus Numerantium XXI, Utilitas Math, 1978), pages 307-321. M. Hall and L.J. Paige, Complete mappings of finite groups, Pacific J . Math., 5 (1955), 541-549. A.D. Keedwell, On orthogonal latin squares and a class of neofields, Rend. Mat. (Roma) ( 5 ) 25 (1966), 519-561. A.D. Keedwell, On the sequenceability of non-abelian groups of order pq, Discrete :?lath. , 37 ( 1981 ), 203-216. L.J. Paige, A note on finite abelian groups, BUZZ. h e r . Math. Soc., 53 (1 947), 590-593. L.J. Paige, Complete mappings of finite groups, P a c i f i c J . Math., 1 (1951), 111-116.
Department of Mathematics , University of Surrey, Gu i 1 dford , Surrey, GU2 5XH, England
535
ON R-SE()UENCEABILITY AND Rh-SEQUENCEABILITY OF GROUPS A.D.
Keedwell
ABSTRACT A f i n i t e group (G,.)
o f o r d e r n i s s a i d t o be R-sequenceable ( o r near-se-
quenceable) i f i t s elements
a o y aly
..., a n-1
t h e p a r t i a l p r o d u c t s bo = a o y bl = aoaly
can be o r d e r e d i n such a way t h a t
...,bn-2
b2 = aoala 2y...y
a r e a l l d i f f e r e n t and so t h a t t h e p r o d u c t bn-l
=
aoala2...an-l
= a a a
= b
0
o 1 2**'an-2 We show
= e.
t h a t a f i n i t e d i h e d r a l group i s R-sequenceable i f and o n l y i f i t i s o f d o u b l y even o r d e r and t h a t a n o n - a b e l i a n group o f o r d e r pq such t h a t p has 2 as a p r i m i t i v e r o o t , where p and q a r e d i s t i n c t odd primes w i t h p
< q, i s R-sequenceable.
We also
d i s c u s s t h e g e n e r a l i z e d concept o f R h - s e q u e n c e a b i l i t y and i t s r e l e v a n c e t o t h e c o n s t r u c t i o n o f s e t s o f m u t u a l l y o r t h o g o n a l l a t i n squares.
1. INTRODUCTION We b e g i n w i t h some d e f i n i t i o n s .
A f i n i t e group (G,.) o f o r d e r n i s s a i d t o be sequenceable i f i t s elements e, aly
can be a r r a n g e d i n a sequence a.
a2,
..., a n-1
i n such a way t h a t t h e
are a l l = a a a n-1 o 1 2**'an-1 d i s t i n c t (and consequently a r e t h e elements o f G i n a new o r d e r ) . I t i s s a i d t o p a r t i a l p r o d u c t s bo = a o y bl = aoaly
b 2 = aoala 2y...yb
be R-sequenceable (see [ 21) o r near-sequenceabk
(see [ 51) i f i t s elements can be
o r d e r e d i n such a way t h a t t h e p a r t i a l p r o d u c t s bo = a o y bl = aoaly b
2 bn-l
a r e a l l d i f f e r e n t and so t h a t t h e p r o d u c t o 1 2.~e~n-2 = a a a = b =e. o o 1 2***an-1 A one-to-one mapping g + O ( g ) o f a f i n i t e group (G,.) o n t o i t s e l f i s s a i d t o
= aoa,a 2y...yb
n-2
= a a a
be a compZete mapping i f t h e mapping g
+
$ ( g ) , where $ ( g ) = g . o ( g ) i s a g a i n a one-
I f (Gym) i s R-sequenceable, t h e n t h e mapping
to-one mapping o f G o n t o i t s e l f . -1 f o r i = 1,2,...,n-2,e(bn-l) = al&(c) = a o ( b i ) = b i. b i+l i+l t h e element o f G which does n o t o c c u r i n t h e s e t {bo,bl,b2,...,b
= a
0
n-1
= e, where c i s
1, i s a com-
536
A.D. Keedwell
p l e t e mapping, as was shown by
L.J. Paige i n [ 7 ] .
Thus t h e c o n d i t i o n t h a t (G,.)
be R-sequenceable i s a s u f f i c i e n t c o n d i t i o n f o r G t o have a complete mapping b u t i t i s n o t always necessary. btl-l
For example, f o r a b e l i a n groups, t h e c o n d i t i o n
= e i s alone s u f f i c i e n t (L.J.
Paige [6]). When and o n l y when a group G has a
complete mapping, t h e l a t i n square formed by the Cayley t a b l e o f G has an orthogonal mate (see s e c t i o n 1.4 o f [ 1 ] f o r a p r o o f ) and, i f t h e group i s R-sequenceable, t h i s orthogonal mate can be constructed by t h e procedure c a l l e d t h e c o l m r*:=;rm> i n [ 4 ] .
(See a l s o s e c t i o n 7.4 o f [ 11).
The column method makes use o f the
permutation 6 d e f i n e d above which we can w r i t e i n c y c l e form as ( c ) ( b l b2 b3 -1 -1 -1 -1 bn-l) and f o r which t h e elements bl b2, b2 b3, bn-2bn-1, bn-lbl a r e the
...,
-1 -1 I f f u r t h e r bl b3, b2 b4,
...
-1 ..., bn-2bl,
b - l b i s again n-1 2 a r e - o r d e r i n g o f t h e n o n - i d e n t i t y elements o f G, then t h e column method permits n o n - i d e n t i t y elements o f G.
t h e c o n s t r u c t i o n o f a t l e a s t t h r e e m u t u a l l y orthogonal l a t i n squares based on the Cayley t a b l e o f G. More g e n e r a l l y , we s h a l l say t h a t a group (G;) a!Zt
o f order n i s Rh-scquenca-
i f n-1 o f i t s elements can be arranged i n a sequence cl,
c2,
..., c n-1
in
such a way t h a t the s e t o f elements c-’ciil f o r i = 1,2,...,n-l, are a l l d i s t i n c t i (where a r i t h m e t i c o f s u f f i c e s i s modulo n-1), and l i k e w i s e the s e t s o f elements -1 -1 -1 c . c i + 2 ’ c . Ci+3’ c 1. c i + h ’ I n p a r t i c u l a r , a group which i s R 1-sequenceable 1 i s R-sequenceable w i t h -1 -1 -1 -1 = c c a = e and al = c ~ - ~ ca2~ =, c c 2 , a3 = c2 c3, a 0 n-1 n-2 n-1’ I f a group ( G , . ) i s Rh-sequenceable, the column method permits t h e c o n s t r u c t i o n o f
...,
,
..., ...@
a t least h
+ 1
m u t u a l l y orthogonal l a t i n squares based on t h e Cayley t a b l e o f G
( a s has been shown i n [ 41 and [ 1 1 ) . R. Friedlander, 8. Gordon and M.D. t l i l l e r have i n v e s t i g a t e d the R-sequencea b i l i t y o f various classes o f a b e l i a n groups i n [ 2 ] .
Here we o b t a i n some r e s u l t s
on the R-sequenceability o f c e r t a i n non-abelian groups and make a few observations about Rh-sequenceability. I n t h e i n v e s t i g a t i o n o f s e q u e n c e a b i l i t y o f groups, t h e concept o f a quotient ne~7~lenoing has played an i m p o r t a n t r o l e .
We s h a l l make use o f t h e same concept i n
t h e present paper.
DEFINITION: I f
a i s any sequence u
, u2,
..., un o f elements o f a group G,
P(.x) w i l l denote i t s sequence o f p a r t i a l products v1 = ul,
v2 = u1u2,
then
531
On R-sequenceability and Rh-sequenceability of groups v
= u u u3 a . .
.,..., vn = u1u2.. .u n'
L e t G be a non-abelian group o f order n
which has a normal subgroup H o f order w, and l e t G/H = gp I X ~ , Xwhere ~,...,X~} t = n/w.
A sequence a o f length n c o n s i s t i n g o f elements o f G/H i s c a l l e d a
quotient sequencing o f G i f each x
i' The image under the mapping $:G
1
Q
-+
i Q t, occurs w times i n both a and P(a).
G/H o f e i t h e r a sequencing o r a near-se-
quencing o f G i s a q u o t i e n t sequencing o f G.
I n the l a t t e r case, the l a s t e n t r y
i n t h e q u o t i e n t sequencing i s necessarily the i d e n t i t y element o f G/H.
2. R-SEQUENCEABILITY OF DIHEDRAL GROUPS n 2 The dihedral group On = gpCa,8 : a = E , B = E , aB =
Ba
-1
1 o f order 2n cannot
be R-sequenceable i f n i s odd since the product o f a l l i t s elements takes the form n s -1 B CY f o r some i n t e g e r s , 0 Q s < n, when the r e l a t i o n a6 = @a i s used t o s i m p l i f y i t and t h i s cannot be equal t o the i d e n t i t y element unless n i s an even integer.
We have been able t o show t h a t t h i s necessary c o n d i t i o n f o r R-sequence-
a b i l i t y i s also sufficient.
That i s , the dihedral group Dn o f order 2n i s R-se-
quenceable i f and only i f n i s an even integer. quencing i n FIG. 1.
We e x h i b i t a s u i t a b l e R-se-
This i s based on the q u o t i e n t sequencing
1 1 . . . 1 m
x
1 1 . . . 1
1's
m
1's
x
x . . . x
(2m-1)
x's
n 2 where n = 2m, H = gpCa : a = € l a n d D /H = { l a x ] . The cases n = 4h and n = 4h n have t o be t r e a t e d separately. (The R-sequenceability o f Dn f o r the p a r t i c u l a r
-
values n = 2,4,6,8,10,14,20,22,32,34,36,38,50,64,66
and f o r eighteen other values
between n = 100 and n = 4084 has already been established i n [ 211 .)
3.
R-SEQUENCEABILITY OF NON-ABELIAN GROUPS OF ORDER pq
I n a recent paper [ 5 1 , we showed t h a t the non-abelian group o f order pq, where p and q are d i s t i n c t primes w i t h p prime which has 2 as a p r i m i t i v e element. also R-sequenceable.
< q, i s sequenceable whenever p i s a We show next t h a t these same groups are
Our method i s almost a copy o f t h a t described i n [ 51 and
uses the same q u o t i e n t sequencing. L e t H be the unique normal subgroup o f order q i n the given group G o f order pq and suppose t h a t the f a c t o r group G/H i s generated by the element x.
A neces-
A.D. KeedweN
538
+
sary c o n d i t i o n f o r t h e existence o f G i s t h a t q = 1 i n t e g e r h.
2ph f o r some p o s i t i v e
We suppose t h a t p has 2 as a p r i m i t i v e element and l e t u s a t i s f y t h e
congruence 20. h oh 5 1 * 2
= 1 mod p, f
1 mod p.
Then o i s a p r i m i t i v e element o f GF[p] s i n c e The f o l l o w i n g i s a q u o t i e n t sequencing f o r G (as i s
shown i n [ 51):
a sequence o f 2ph l ' s , f o l l o w e d by x, f o l l o w e d by a sequence o f 2ph-1 p-2 p-3 p-2 2 3 2 0-1 5 -u 0 - 0 0 -u 1-0 (p-1)-tuples x x X x X , where i n d i c e s a r e computed
...
modulo p, f o l l o w e d by a sequence 0-1 xu2-a 03-02 ap-2-0p-3 X X x
...
1 x
2
x
4
x
8
... x 2P-2
l-oP-2 X
2 xu Jp-3
*
The p a r t i a l products a r e 2ph l ' s , f o l l o w e d by the sequence x x' 2 p-2 p-2 repeated 2ph-1 times, and then the sequence x x' xu XU x' 2 x' xu x 1. (We use the f a c t t h a t o ~ = - 2.)~
...
X
2-2
... x
p-4
X0
...
L e t G be generated by two elements a, b such t h a t a' ab = ba
S
, where
sp
9
= e, bp = e and
1 mod q and e i s t h e i d e n t i t y element o f G.
Then,
X
(b'av)(bxaY)
u+x vs +y a and G has a unique normal subgroup H = gpIa : aq = e l o f
= b
The n a t u r a l homomorphism G 2 elements 1 = H, x = bH, x2 = b H,
order q.
+
G/H maps G o n t o t h e c y c l i c group C
..., xP-'
= bP-lH.
P
with
L e t r be a p r i m i t i v e element o f GF[q]. Then (compare [ 5 1 ) we seek a nearh-2 ,h-3 h h-1 ,h+l-,h r ,a , a 9 by sequencing o f G i n t h e form e, ar
-'
11
= r
...,
h-1- h-2 r . We make use o f t h e f a c t s t h a t oP-r
t h a t ap-r-Gp-r-l
= -oP-r
, where
2a
= Zr-',
t h a t l-oP-'
= -1,
and, generally,
1 mod p.
This sequencing g i v e s r i s e t o t h e p a r t i a l products which a r e l i s t e d i n vw FIG. 2 . (For ease o f p r i n t i n g , we have represented t h e element b a by the The expressions denoted by E ( j ) ( f o r i and j = O,l,...,p-2) i n FIG. 2 a r e as given below. We equate
ordered p a i r v, w i n FIG. 2.)
i = 1,2,...,q-2
these expressions t o expressions i n a p r i m i t i v e element t o f G R q ] e x a c t l y as i n
I 5 I.
We a l s o make t h e d e f i n i t i o n s a!?! = t a ( J ) f o r i = l,Z, ...,qi
2 and
5 39
On Rsequenceability and Rh-sequenceability of groups
... +...+a
+a
it1 U
= t
p-2
it1
Q 1-1 i-1 ( t -t )t
.
From these e q u a l i t i e s , we o b t a i n E(l)-E(o)su-l 1 1
U
-a2
= ( t l-su-l)(ta-ta-l),
1
2
u u E(2)-E(1)su -a= (2)-a(2) = ( t 2-t '5 1 1 a2 1
2
-a)(te-tL-l),
... E (P-2) -E(P-3 1 1
p-2
Iso-a
tE(o)-E(P-2)s1-o 1 1
P-2
p-3
(P-2)-a(P-2) = a2 1
-
t a ( P - ' )-a(P-l) 1 1 U
Hence, a") 1
= t"-'(t
U
= ( t P-2-t up-3 ap-2-up-3 U
= ( t - t p-2S1-ap-2
Q a-1 ) ( t -t
L 11-1 ) ( t -t ) Y
1.
1 0-1 -s ),
( 2 ) = t a - l ( t u2 -t o2-0 al s ), .(3) 1
u u 3 2 ( t 3-t 2s u -a 1,
=
...
(P.2) al
=
U
P-2-t
u P-2-aP-3 P-3Sa
) Y
U
(P-1) = ta-l (t-t al
I t i s easy t o check, using FIG. 2 , that, w i t h these choices f o r the expres-
sions E(k) and the indices a(') i i
i n terms o f the p r i m i t i v e element t o f GF[q], a l l
5 40
A.D. Keedwell
t h e p a r t i a l products o f the form b'
j
(j) aB1 a r e d i f f e r e n t i n c l u d i n g t h e l a s t one
U
which occurs provided t h a t
p =
( T h i s has been i l l u s t r a t e d i n [ 51).
t p-2(-te-2)).
F i n a l l y , i t i s easy t o see t h a t a l l b u t the l a s t o f t h e p a r t i a l products We r e q u i r e i n a d d i t i o n t h a t t h e l a s t
which a r e powers o f a alone a r e d i f f e r e n t .
such product i s equal t o t h e i d e n t i t y element e. -I:
t
-2
p l ) ( l - s - l ) + LIS 9- 1
That i s we r e q u i r e U
0 mod q.
u
Using t h e f a c t t h a t
= t p - 2 ( - t e - 2 ) and
U
that
= t'-'(t-t
9- 1
'-'s-'),
U
t h i s becomes
U
t P-2te-2ttL-2(t-t p-2s-1)(i-s-1)
Since u = r
h-1
-r
-
t
Up-2 1-2
t
s
-2-
- 0, whence t U p-2-1 = -1. U
h-2 . h-1 ,h-2 i n FIG. 2, we must a l s o have r
-
= t p-2(-te-2).
That
We conclude t h a t an R-sequencing (near-sequencing) o f t h e non-abelian group o f order pq, where 2 i s a p r i m i t i v e element o f GF[pl, can be obtained by t h e f o l l o w i n g procedure. F i r s t choose one o f t h e r o o t s s o f t h e congruence sp compute the i n t e g e r o such t h a t 20
=
1 mod q and a l s o
Then choose p r i m i t i v e elements r up-2-1 such t h a t , modulo q, t = -1 and and t of GFlqI and i n t e g e r s u1,u2,...,u U u -u 2 u -u y 2 2 U aP-2-uP-3 1 so-l 2 1 3 2 su - u p 2-'p -3 s a -u , t t # s ( s i n c e we
.
1 mod p.
,..., -
~
r e q u i r e a ( J ) # 0 f o r j = 1,2, ...,p- 1 ) . Compute the i n d i c e s a ( J ) f o r i 1 9,-1 i = 1,2 q-1 and j = 1,2 p-1. F i n a l l y choose h so t h a t r h - 2 ( r - l ) = t
,...,
modulo q .
,...,
This gives an R-sequencing f o r a r b i t r a r y choice o f t h e index 9,.
I n order t o i l l u s t r a t e t h e procedure, we have c a r r i e d o u t t h e necessary c a l c u l a t i o n s f o r the case p = 5 and q = 11 and we d i s p l a y t h e r e s u l t i n g nearsequencing o f t h e non-abelian group o f order 55 i n FIG. 3. Since 20
1 mod p, we have u = 3.
We may choose s = 3 ( s i n c e 35
and r = t = 2 ( s i n c e 2 i s a p r i m i t i v e element o f GFI111).
t5
-1 mod 11 so u
3
= 6.
u u zero ifwe choose t ' = t
We f i n d t h a t a \ ' ) , = 1.
We a l s o p u t t
a{3'
!L-1
= 1.
1 mod 11)
Then (and
al( 4 ) )
are a l l non-
Then u = 1 = 2
h-2
(2-1),
so
541
On R-sequenceability and Rh-sequenceabifity of groups U
h = 2.
We o b t a i n
1 0-1 = 1 -s
a:’)
= t
.(2) 1
= t 2-t
u
(4) = t .l
-
u
2 $0
-
3
-u= 1
2
-
= 3, 36 = 9 ,
u 3 t 3 s 1-0 = t ( l + s -1 ) = 2(1+4) = -1,
where these l a t t e r c a l c u l a t i o n s a r e made modulo 11. We a l s o have t h e group 2 2 9 4 4 4 g e n e r a t i n g r e l a t i o n s ab = ba3, ab = b a , ab3 = b3a5, ab = b a
.
4. R,-SEQUENCEABILITY I n g e n e r a l , t h e problem of d e t e r m i n i n g which groups a r e Rh-sequenceable seems t o be a d i f f i c u l t one.
F o r a b e l i a n groups G, i t i s c e r t a i n l y necessary
t h a t t h e Sylow 2-subgroup o f G s h o u l d be e i t h e r t r i v i a l o r n o n - c y c l i c . However, even f o r c y c l i c groups, t h e problem i s n o t t r i v i a l . group i s R-sequenceable i f and o n l y i f i t has odd o r d e r . G. R i n g e l and a l s o has been proved i n 121. C4mtl
2m-3,
Certainly a cyclic
T h i s has been s t a t e d by
The f o l l o w i n g R-sequencings for
and C4mt3
are a l t e r n a t i v e t o those given i n t h e l a t t e r reference.
F o r C4m+l
an R-sequencing i s
...,..., 2m-2,
0 4m, 2, 4m-2, 4, 4m-4, 6, 4m-6, 2mt5,
...,..., 2m-(2m-3),
F o r C4m+3
2m+(2m-1),
(See [ 3 ] . )
4m-(2m-2),
2m-(Zm-l),
2m, 2m-1, 2mt3,
2m+l.
an R-sequencing i s
0, 4mt2, 2, 4m, 4, 4m-2, 6, 4m-4, 2m-1, 2m+5, 2m-3,
...,..., 2m-2,
...,..., 2m-(2m-3),
4m-(2m-4),
2mt(2m+l), 2m-(2m-l),
Zm, 2m+l, 2m+3, 2mt2.
However, a computer search has shown t h a t C9 i s n o t R2-sequenceable,
A
s i m i l a r search has shown t h a t C15 i s n o t R3-sequenceable b u t t h a t i t has 32 i s o m o r p h i c a l l y d i s t i n c t R -sequencings. The c y c l i c group CPl i s R2-sequenceable b u t 2 a t t h e p r e s e n t t i m e t h e q u e s t i o n whether i t i s a l s o R3-sequenceable remains undecided. The R h - s e q u e n c e a b i l i t y o f c y c l i c groups o f p r i m e o r d e r i s covered by t h e f o l l o w i n g theorem: n THEOREM: An eZernentary abeZian group of order p i s Rh-sequemeable for
5 42
A.D. Keedwell
...,p n -2.
h = 1, 2,
i s a p r i m i t i v e element o f t h e G a l o i s F i e l d GF[pn]
PROOF: I f
, the
f o l l o w i n g se-
quencing o f t h e elements o f t h e a d d i t i v e group GF[pn] ( w h i c h i s e l e m e n t a r y a b e l i a n ) has t h e r e q u i r e d p r o p e r t i e s :
0 , w-1,
UJ
2
3
2
u -w
-w,
,
...,
w
n p -2
(The n-1 elements c a r p t h e n 0, ~
1
n
p -3
-0
, 1-w pn-2 .
given i n t h e d e f i n i t i o n o f Rh-sequenceability n 2 o r 1, UI, w , , up -2.)
...
,
T h i s g i v e s a t once t h e well-known r e s u l t t h a t n
n
COROLLARY: r'rotr an e?smentary abelian group of order p a s e t o f p -1 w t u u l l y staz2ogonal lati.? squarss all Lased on t h a t group em be constructed and hence a n
;csai2juaotan pv,it.cti20 plans o f order p ,
EXAMPLE: R -szqusnceatiLLty of C 7 3
c3'
2 8 I n GFI 91, t h e r e e x i s t s a p r i m i t i v e element w such t h a t o = w + 1, w = 1. 3 We can w r i t e C3 x C 3 = { a : a3 = e l x ;b : b = e l o r , i n a d d i t i v e n o t a t i o n , as
i
io,1,-1
! + I
I.
{O,b,VJ
Our R -sequencing i s
7
0, w-1,
w
2
-UJ,
o r 0, 0-1,
1,
3 lL
2
-w
,
O, w t l ,
4
I)
-w
3
, w 5-w 4 , w 6 -w 5 , w 7 -w6
- ~ + l ,-1,
-w,
y
1-w
7
- ~ l .
I n m u l t i p l i c a t i v e n o t a t i o n , we have t h e sequencing 2 2 2 2 2 e , a b, a, b, ab, ab2, a2, b , a b w i t h p a r t i a l p r o d u c t s e, a2b, b y b2, a, a b , 2
n
ab
L
, ab,
e.
Thus t h e sequence o f n-1 elements c
1 ' c2; d e f i n i t i o n o f R h - s e q u e n c e a b i l i t y becomes e, a2b, b, b group.
..., c n-1
, a,
given i n the
a2b2, ab2, ab f o r t h i s
The f i r s t " d i f f e r e n c e s " c f ' ci+l a r e as l i s t e d i n t h e sequencing above. -1 2 2 2 2 The second " d i f f e r e n c e s " c . a r e b, ab, ab , a2, b2, a b , a b, a; and so on. 1
5 43
On R-sequenceability and Rh-sequenceability of groups 4h-2
e a
2h-1
e
e a
2h-1
- ( 2h-4)
a
a
a
a a
2e-1 a -(2a-2) a
a
a
-2 a
a B
a
2h-2
h+k-1
a
- ( 211-3) a
2 a
h
a
-1 a
3h-1
a
3h+l
a
3h-2
a
Ba
6a
-2 Ba Ba
Ba
-(2e-2)
3h-2 3h-1 3h-3 3h
3h+t-3 3h-9.-1
Ba
a
22-1 Ba
3htt-2
2h
-(2h-1) -1
Ba
3
3h+.t-1
Ba
h-1
Ba
3h-a+l
3h-n
h-k+l
4h-3
Ba
Bcr
h+e-2
h a
3h
2a-2
2
h- 2
h-1
f3a
2h-2
21-2 a
h+l
Ba
2h-3
a
h-k+l
Ba
a
a
h-.t
h-E
2
-(2a-1)
2h-2
a
2h-4 a
a
Ba
a
a
-(2h-5)
Ba
-3
a
2
-1
a
e 2h-2
a
a
a
I
- ( 2h-3)
-(2h-2) 2h-3
P a r t i a l products
Ba
2h-3
a
FIG. 1 (continued over.)
4h-3
Ba
A .D.Keedwelt
544 R-sequencing o f D 4h 0
J
P a r t i a l products
-(2h-1) 6a
-2h
2h
B.1
2h
2h 6a
2h+l 6a
2ht2
ii
ea
4h-2
3
2h+l
6a
6a
h
Ba
5a
a
2 5a 3
a.1
3h+l
4h-5
e
3h-3 h-1
a
h
Ba
2h- 1
2h
3h-1
B
6a
2
3h
4h-3
4h-1
5a
Ba
%a
3h+2
2h-1 6a
2h-1
a
4h-4
a
a
2h-2 6s
5
a
6a
a
2h-1 5a
a
h+l
5a
P a r t i a l products
h-2 6a
3h-1
4h-1
6s
i3a
2h+l
h- 1
a
BCI
2h- 1
2h-1 6a
a
En
- (2h-2)
a a
6
2h
?-sequencing o f D 4h-2
2h-3
5a
2h-2 %a
FIG. 1 (continued)
2h-2
%a
e
On R-sequenceability and Rh-sequenceability of groups
0, 0
h
0, r -r 0, r
h-1
h t l rh
-
0, r h-2 - ,h-l
-usa
!J-2
_= -
u
(l)sap-*-a
+a1
(24u +al
p-2
-a
2 t. * .+a
(P-34a 1
p-2
FIG. 2 (contd. on next p a g e )
-0
p-3
(P-2)
+a 1
5 45
5 46
A.D. Keedwell
+.
p-2 p-3 (P-2)+qc2 ..+a 1(P-3)su -' +al i=l
f . . .+a (P-3)s0 1
p-2
p-3
E(Pe2) i
+,(p-')+qi2 E(P-2)+,, 1 i=li P-2
+...+a
1
+...+a
1 P-3 +a
FIG. 2 ( c o n t d )
On Rsequenceability and Rh-sequenceabiHty of groups
Sequencing
7 e 2
a a a a
4
8
5
a
a a a a
10 9 7 3 6
b 2 3 b a 9 ba 3 4 b a 4 10 b a 2 6 b a 7 ba 3 8 b a 4 9 b a 2 b a 3 ba 3 5 b a 4 7 b a 2 2 b a 6 ba 3 10 b a 4 3 b a 2 4 b a ba 3 9 b a 4 6 b a 2 a b a 2 ba
e 2
a
6
a
3
a a
8
a a
7 5
a a a
4 10
baa 3 9 b a 4 3 b a 2 a b a 9 ba 3 10 b a 4 4 b a 2 6 b a b 3 b a 4 6 b a 2 2 b a 4 ba 3 5 b a 4 10 b a 2 5 b a ba 3 2 b a 4 7 b a b2 6 ba 3 7 b a 4 b a
Sequencing
P a r t i a l products
(contd) 3 7 b a 4 b a 2 5 b a 4 ba 3 3 b a 4 2 b a 2 10 b a 8 ba 3 6 b a 4 4 b a 2 9 b a 5 ba 3 b a
(contd)
2 b a 5 ba 3 6 b a b4 2 3 b a 3 ba 3 4 b a 4 9 b a 2 7 b a
ba
10
b3 4 5 b a 2 4 b a
4 8
2 ba 3 3 b a 4 8 b a 2 9 b a 2 10 b a 4 2 b a 3 8 b a 7 ba
b a 2 7 b a 10 ba 3 2 b a a
b2 b4 b3 4 5 b a
e
FIG. 3
5 41
A.D. Keedwell
548
BIBLIOGRAPHY 1. 2.
3.
4. 5.
6. 7.
J. Denes and A.D. Keedwell, "Latin Squares and their Applications". (Akademiai Kiad6, Budapest/English Universities Press, London/ Academic Press, New York, 1974.) R.J. Friedlander, 6. Gordon and M.D. Miller, On a group sequencing problem of Ringel. Proc. Ninth S.E. Conf. on Combinatorics, Graph Theory and Computing. Florida Atlantic Univ., Boca Raton, 1978. (Congressus Numerantium XXI, Utilitas Math, 1978), pages 307-321. M. Hall and L.J. Paige, Complete mappings of finite groups, Pacific J . Math., 5 (1955), 541-549. A.D. Keedwell, On orthogonal latin squares and a class of neofields, Rend. Mat. (Roma) ( 5 ) 25 (1966), 519-561. A.D. Keedwell, On the sequenceability of non-abelian groups of order pq, Discrete :?lath. , 37 ( 1981 ), 203-216. L.J. Paige, A note on finite abelian groups, BUZZ. h e r . Math. Soc., 53 (1 947), 590-593. L.J. Paige, Complete mappings of finite groups, P a c i f i c J . Math., 1 (1951), 111-116.
Department of Mathematics , University of Surrey, Gu i 1 dford , Surrey, GU2 5XH, England