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Variations on a scheme of McFarland for noncyclic difference sets
JOURNAL
OF COMBINATORIAL
THEORY,
Series A 40, 9-21
(1985)
Variations on a Scheme of McFarland for Noncyclic Difference J. F.
Fort
DILLON
Department of Defense, George G. Meade. Maryland
Communicated
by William
Received
1.
Sets
October
20755
M. Kantor
30, 1980
INTRODUCTION
The k-subset D of the group G of order u is called a (u, k, A, n)-difference element of G has exactly 1” representations as a difference of two elements of D; the parameter n is defined to be equal to k - I.. Using multiplicative notation for the group operation, we have that D is a difference set precisely when it satisfies in the group ring Z[G] the equation
set if every nonidentity
DD'~-"=n l,+%G,
(1.1)
where D'- ‘) denotes the set of the inverses of elements of D and 1, denotes the identity element of G. McFarland [3] has given a construction for difference sets having the parameters
(1.2)
where q is any prime power and s is any positive integer. These difference sets are in the groups of the form
G=ExK, 9 0097-3165185
$3.00
Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
10
J. F. DILLON
where E is the elementary abelian group of order qs+’ and K is any group of order r+l, r=(q’+’ - 1)/( q - 1). In particular, this construction produces Hadamard difference sets in any group of order 22(s+‘) which contains Z;+ ’ as a direct factor. In this brief paper we point out two variations on McFarland’s scheme; the first extends the construction to a much larger class of groups and the second produces many more inequivalent difference sets in some of the same groups. In Section 2 we show that the elementary group E need not be a direct factor of G; indeed, essentially the same construction produces difference sets with parameters (1.2) in any group G of order u whose center contains E. In particular, G may be any abelian group of order u which contains E. Taking q = 2, we thus obtain Hadamard difference sets in all groups z; x zp+2.
These difference sets (in abelian groups of order 22s+2 and exponent 2s+2) demonstrate the sharpness of the exponent bound established by Turyn [6] in 1965. Turyn himself gave examples of such difference sets in the case s = 1; our construction produces the first known examples of such difference sets for all s > 1. We present these and other general results for 2groups in Section 3. In Section 4 we restrict our attention to groups of order 16 (the case s = 1) showing that the basic construction may be extended to groups which contain E = Z, x Z, as a normal subgroup. This general construction produces difference sets in ten of the fourteen groups of order 16. We complete this section by showing that the cyclic and dihedral groups are the only groups of order 16 which do not contain a nontrivial difference set. The last section presents our second variation of the basic construction which applies when s = 1 and q is not a prime. Whereas McFarland’s construction employs the l-dimensional subspaces of E regarded as a 2-dimensional GF(q)-linear space, we observe that any spread (i.e., family of q + 1 subgroups of order q the intersection of any two of which is the identity) for E will do as well. Furthermore, in the groups G = E x K, q odd, difference sets produced from inequivalent spreads are themselves inequivalent so that McFarland’s lower bound of (q + 1)/2 for the number of inequivalent difference sets in G may be multiplied by the number of inequivalent spreads for E. We also point out that in the Desarguesian case considered by McFarland his lower bound may be improved to q - 1 and that even this is a gross underestimate for most q. The details of the more general construction are given in the next section. We employ, for the most part, the terminology and notation of McFarland [3].
NONCYCLIC
DIFFERENCE
11
SETS
2. THE CONSTRUCTION Let q be a prime power and let s be a positive integer. Let E be the elementary abelian group of order qsfl which we regard as a vector space of dimension s + 1 over the finite field F= GF(q). Let H,, HI,..., H,, r = (4‘+ ’ - l)/(q - l), be the hyperplanes (i.e., s-dimensional subspaces) of E. Since every nonidentity element of E is contained in precisely (q’ - l)/ (q - 1) hyperplanes we have in the group ring Z[E],
i;, H,=rl.+
{(q”- l)/(q-
1)) (E- 1~).
(2.1)
Let G be any group of order u = qs+ ‘(r + 1) which contains E as a normal subgroup and let G=g,E+g,E+
... +g,+lE
be the coset decomposition of G with respect to E; i.e., {g,, gz,..., gl+, } is any set of representatives of the cosets of E in G. If we regard the cosets g,E as being the elements of the factor group G/E, then the elements
g,E, g,&., g,E constitute a (trivial) (r + 1, r, r - 1, 1)-difference set in G/E. Equivalently, in the group ring Z[G] we have i g,g;‘E=E+(r-l)G=rE+(r-l)(G-E). i.i= 1
(2.2)
Now let D be the subset of G given by D=g,H,+g,H,+
... +g,H,,
so that DD’-“=
i i,j=
g,H,H,g;’ 1
=q’ i
giHig,:‘+q”-’
i=l
i i,j=
i #j
g,g;‘E. 1
By (2.2) the second of these sums is (r - l)(G - E). Thus, D is a difference set in G if and only if qs 1 giHig;’ ,=I
=q”rl.+q”-‘(r-
1) E*,
12
J. F. DILLON
where E* denotes the set of nonidentity elements q(q” - l)/(q - l), this condition is equivalent to ;$I g,H,g;’
=rl,+s
of E. Since r - 1 =
E*;
i.e., every nonidentity element of E must belong to exactly (q’ - l)/(q - 1) of the subgroups gjHig; ‘. Of course, this condition is guaranteed (indeed reduces to (2.1)) if the map Hi --+g, H,g; ’
happens to be a permutation foregoing in the
of the hyperplanes of E. We summarize
the
THEOREM. Let q be a prime power and let s be any positive integer. Let E be the elementary abelian group of order qS+I (which we regard as a vector space of dimension s + 1 over GF(q)), and let H, , Hz,..., H,, r= (q’+’ - l)/(q - l), be the s-dimensional subspacesof E. Let G be a group of order q”+ ‘(r + 1) which contains E as a normal subgroup and let g,, g2,..., g, be elements of G lying in distinct cosets of E. Let D=g,H, +g,H,+ “’ + g,H,. If the coset representatives gi should have the property that { giHTg,: I : 1 < i < r} is a l-design on the point set E*, then D is a difference set with parameters (1.2). COROLLARY. If the subgroup E of G lies in the center of G, then D is a difference set for all choices of the coset representatives g,, g2,..., g,.
McFarland’s result in which E is assumed to be a direct factor of G is, of course, a special case of this corollary. We observe that when q is a prime the family (Hj} of hyperplanes of E is invariant under conjugation by any element of G and if g,, gZ,..., g, are elements of G with no two in the same coset of E then the r x r array A = W$g,?),
1 Q i,j< r,
has the property that every row is a permutation then clear.
of {Hi}.
The following is
COROLLARY. The group G contains a nontrivial difference set if the corresponding array A has a transversal; i.e., there exists a permutation z of { 1, 2, 3,..., r} such that the r entries {A(i, z(i)): 1 < i < r> are distinct.
We shall have occasion to use this last result in Section 4 where we determine the groups of order 16 which contain difference sets.
13
NONCYCLIC DIFFERENCE SETS 3. DIFFERENCE SETS IN ~-GROUPS
The results of the previous section are particularly special case q = 2 when the parameters become
interesting
(u, k, A, n) = (z2”+ 2, z2’+ 1- 2”, z2” - 2”, z2”).
in the (3.1)
Mann has shown [2] that any nontrivial difference set in a a-group (indeed, any SBIBD having u a power of 2) must (up to complementation) have parameters of the form (3.1). Since these parameters satisfy the relation v=4n.
any difference set in a 2-group is necessarily a Hadamard difference called because the incidence matrix of the associated block design, O’s and l’s replaced by l’s and - l’s, respectively, is a Hadamard Indeed, if we associate with any element D in Z[G] the D* = G - 20, we see that the equation DD”‘=nl
is equivalent
G
set-so with its matrix. element
+r”G *
to D*D*’
“=4nl,
+ (v-412) G.
which implies the previous remark. Thus, in a 2-group G of order v = 22sc2 a subset D is a difference set precisely when its associate D* satisfies D*D*‘-‘I=
ICI 1,.
Let &’ denote the class of all finite groups which contain a Hadamard difference set. X contains the group of order 1 and both groups of order 4 the difference sets in which are singletons and their complements. It is well known that Y is closed under the taking of direct products; indeed, if G is the (not necessarily direct) product of two “disjoint” subgroups G, and G, containing Hadamard difference sets D, and D, respectively, then D = D, (G, - D,) + (G, - 0,) D, has associate D* = DTD,* which satisfies (DTDT)(D:‘-“Dr~“)=DTIGzI
lczDT’-“=
IG, I l,, lG21 lGz= ICI 1,.
We thus have the Remark. The class X’ of groups which contain a Hadamard difference set is closed under arbitrary products; i.e., if G = G, G, with G, and G, in 2 and IG, n G, I = 1, then G also belongs to &‘. Since X contains the groups of order 4 we have the
14
J. F. DILLON
COROLLARY. (Turyn and Kesava Menon). is a direct product of groups of order 4.
In his remarkable
2 contains any group which
paper [6] of 1965 Turyn established the
THEOREM. If the abelian group G of order 22S+2 belongs to &‘, then G has exponent at most 2” + ‘.
Turyn himself [6] gave examples of such difference sets in the group Z2 x Z, (i.e., the case s = 1) and asked (private communication) if his exponent bound was sharp for larger s. McFarland’s construction produces difference sets in Z2+ l x Z=>+I which has exponent 2Sf ‘, within a factor of 2 of Turyn’s bound. Our construction demonstrates the sharpness of Turyn’s bound for all s. The theorem of the previous section takes the following simpler form in the case q = 2. THEOREM. Let E be the elementary abelian group of order 2” + I and let H, , H2 ,..., H,, r = 2”+ ’ - 1, be the subgroups of E of order 2”. Let G be a group of order 22sf2 which contains E as a normal subgroup and let g,, g2,..., g, be elements of G lying in distinct cosets of E. Let D =g, H, + g,Hz + . . . +g, H,. If the coset representatives gi should have the property that {g,H,g;‘:
1 Qi
{Hi: 1
then D is a difference set in G with parameters (3.1). COROLLARY. D is a difference set if E lies in the center of G.
The sharpness of Turyn’s bound is now established by this last corollary which implies the COROLLARY. D is a difference set in the group G=Z;xZ,,+s.
Notice that this group G contains a unique subgroup isomorphic to Z;+ ’ which we take as E. Actually, the factor Z2$+2 of G in this last corollary may be replaced by an arbitrary group of order 2’+=. More generally we have the COROLLARY. Zf the group G of order 22S+2is the direct product of s + 1 nontrivial subgroups, then G contains a Hadamard difference set.
15
NONCYCLICDIFF'ERENCESETS
wherelGil>2foreachi, ldi
which we may take as E in the first corollary to the above theorem.
Q.E.D.
We thus obtain Hadamard difference sets in many different groups of order 22S+2 including many nonabelian groups. We illustrate the construction for the special case
where D2’+2 denotes the dihedral group generated by elements a and b satisfying a2 = 1 = b2’+ ‘, Then 8 = b2’ is an involution the subgroup
of order 2Si2. Let DZE+z be
aba=b-‘.
in the center of DzJ+l and we may take as E
whose cosets in G are represented by the 2’+ ’ elements { 6’: 0 < i < 2”) u {ab’:O
D=
c i=
b’e,H,+ I
23+1-I 1 ab’e,H, I=
2s
is a difference set in G.
4. HADAMARD GROUPSOF ORDER 16
The fourteen groups of order 16 are catalogued in Burnside [l] which also conveniently lists for each group its center. Exactly eight of these groups (including, of course , the four noncyclic abelian groups) have center containing Z2 x Z, which we may take as E to construct difference sets according to our theorem. Of the remaining six groups, exactly two contain a normal Z2 x Z2 which we may take as E to construct difference sets according to our theorem and the following Remark. 582a/40/1-2
Let
G
be
a
group
of
order
16
containing
E=
16
J. F. DILLON
h, > = Z, x Z, as a normal subgroup, and let g,, g,, g, be {l=h,,h,,h2, elements of G which lie in distinct cosets of E. Then the matrix A = [g,h,g;l] 1 < i,j< 3 has a transversal. ProofI E must intersect the center of G so that for some j, the j,th column of A is constant h,. The 3 x 2 submatrix of A comprised of the remaining two columns must then have rows which are permutations of the remaining two h’s and two of these rows must be the same. Then either diagonal constitutes a transversal of this 2 x 2 submatrix; and, taking hjO from the remaining row, we obtain a transversal for all of A. Q.E.D.
Appropriate modification of the above argument the affirmative the first few cases of the following
allows us to answer in
Query. Let G be a group of order 22S+2 containing a normal subgroup E isomorphic to Z;+ ‘. Let Hi, Hz,..., H,, r = 2’+ i - 1, be the hyperplanes 2,..., g, be elements of G lying in distinct cosets of E. Let ofEandletg,,g Let A = (g,H,g,: I), 1 d i,j < r. Must A have a transversal? The four groups of order 16 which do not contain a normal Z, x Z2 are I. II. III. IV. Q-IpQ=p-‘,
G cyclic; i.e., G = (P), PI6 = 1; G dihedral; i.e., G = (P, Q), P8 = 1 = Q2, QPQ = P-‘; G = (P, Q ), P8 = 1 = Q2, QPQ = P3; G generalized quaternion; i.e., G = (P, Q), P8 = I= Q4, Q2=p4.
The following easily verified result shows that groups III and IV, though not yielding to our general theorem still contain difference sets so belong to 2. Remark. Let G be either of the groups given by III and IV. Let D be the subset given by
D = { 1, P2, Q, QP, QP”, QP”}. Then D is a difference set in G.
The cyclic and dihedral groups of order 16 do not contain (nontrivial) difference sets-the cyclic group by Turyn’s exponent theorem and the dihedral group by the following THEOREM. Let H be an abelian group and let G be the generalized dihedral extension of H; i.e., G = (Q, H), Q2 = 1, QhQ = h-’ Vh E H. If G contains a difference set, then so doesevery abelian group which contains H as a subgroup of index 2.
NONCYCLIC
DIFFERENCE
17
SETS
Proof. We have G = H + QH. Let D be a difference set in G which we may write as D=X+QY,
where X and Y are subsets of H. Then DD’-“=
(X+
QY)(X+
QY)‘-“=
(X+ QY)(X’-“+
Y(-“Q)
=Xx’-“+Qyy(~“Q+Qyx’-‘)+xy’-‘)Q =
(xJ+“+
yy(-“1
+2Qyx’-“.
Since D satisfies an equation DD’--“=nl.+AG,
it follows that X and Y must satisfy XX’-“+
YY’-“=nl,+AH; yX’-‘,=dH 2
(*) .
Now let K be any abelian group containing H with [K : H] = 2 and let 8 be any element of K not contained in H. Then K=H+BH.
We define a subset C of K by C=X+BY. Then cc’-‘)=
(X+ eY)(x+ey)'-" = (x+ BY)(X'-"
+ I+“&‘)
=xx(-l’+eyyC-*)e-f
+fjy~(-l)+xy(Gl~~-l
=Xx’-‘)+
yy’-I)+0
{yx’-‘l+e~2(yx(-l))(-l)j
=xX’-“+
yy(-l)+2eypl)
the last equality being a consequence of
y~'-"=~H=e~z(~H)'-"=e-2(y~'~")'-".
18
J. F. DILLON
But then by (*) C satisfies CC’-“=nl,+AH+2e(~H)
2
=nl,+AH+MH
=nl,+X, so that C is indeed a difference set in K.
Q.E.D.
COROLLARY. Zf the cyclic group ZZm does not contain a (nontrivial) ference set, then neither does the dihedral group of order 2m.
dif”
We have thus established the following THEOREM. Of the fourteen groups of order 16 only the cyclic and dihedral groups fail to belong to 2.
5. ANOTHER VARIATION
When s = 1 McFarland’s scheme produces difference sets in the groups of order q2(q+2) of the form G=ExK,
where E is the elementary abelian group of order q2 (regarded as a 2dimensional GF(q)-linear space) and K is any group of order q + 2. These difference sets are of the form
D=g,H,+g,H,+
... +gy+,ffy+l,
where H,, H, ,..., H,, , are the hyperplanes in E and g,, g, ,..., gy+ , are any elements of G lying in distinct cosets of E. That such a set D is a difference set follows (as in Sect. 2) only from the fact that the His constitute a spread for E. We thus have the THEOREM. 5.1. Let q be a prime power and let G = E x K, where E is the elementary abelian group of order q2 and K is an arbitrary group of order be a spread for E and let g,, g2,..., gq+ , be q+2. Let HI, H, ,..., H,,, elements of G lying in distinct cosets of E. Then D = g, H, + is a difference set in G with parameters g,H,+ *.’ +gy+lHy+l (v, k 4 n) = (q2(q + 21, q(q + 11, 4, q2).
19
NONCYCLIC DIFFERENCE SETS
Now suppose that q is odd so that E and K have relatively prime orders and are both invariant under every automorphism of G. Suppose that
and
are difference sets of the type given by the theorem and suppose that D and i.e.,
D’ are equivalent;
for some automorphism
CI of G and some element g of G. Then
1 g;H; = D’ = gD’“’ = c gg; Hia) and, since u fixes E and K, it follows that the two sets {gi} represent exactly the same cosets of E and the image { Hj*)) of a spread for E which must coincide with the spread {Hi.}. spreads (Hi} and (Hi) equivalent if there is an automorphism takes one spread into the other. We thus have the
and {gg;> {Hi} is itself We call the of E which
THEOREM. 5.2. For q odd, inequivalent spreads give rise to inequivalent difference sets.
McFarland’s argument [3] extends to this more general construction to give a lower bound on the number of inequivalent difference sets produced by any fixed spread (Hi}. Let x denote the projection homomorphism fromG=ExKtoEandlet
be any difference set obtained via our construction from the spread {Hi}. Then for each i, 1 < i < q + 1, n(g,) Hi is a coset of Hi in E and the image under z of D, namely,
n(D)=c dir;)Hi, as an element of the group ring Z[E], has coefficient on any element e E E equal to the number of cosets z(gi) Hi containing e. Note that n(D) = 1 n( 8,) Hi may be interpreted
as a “configuration”
in the plane determined
by the
20
J. F. DILLON
spread (the points of this plane are the elements of E and the lines are the cosets in E of the components Hi of the spread). Thus, the coefficient on an element e E E in rc(D) is the number of lines in the configuration incident to e. Clearly the multiset of multiplicities given by the coefficients in the group ring element rc(D) constitutes an invariant for the class of difference sets
equivalent to D. McFarland employed the less discriminating invariant given by the largest coefficient and constructed (q + 1)/2 inequivalent difference sets
by choosing for each such t coset representatives g(lr), gy),..., g!i, property that
with the
n( gj”) E Hi for exactly q + 1 - t values of i, 1 d i 6 q + 1. In geometric terms this means that the configuration rc(D,) has exactly q + 1 - t lines passing through the origin (i.e., containing the identity element of E) and since this number is more than half the number of lines in the configuration no other point can have a coefficient so large. Thus, for each t, q + 1 - t is the largest coefficient occuring in rc(D,) and so the difference sets (0,: 0 < t < (q - 1)/2} are indeed inequivalent. We reiterate the foregoing as the THEOREM 5.3. Let q be an odd prime power and let G = E x K, where E is the elementary abelian group of order q2 and K is an arbitrary group of order q + 2. Let S(q) be the number of inequivalent spreadsfor E. Then G contains at least S(q) x (q -t 1)/2 pairwise inequivalent difference sets.
For the Desarguesian case assumed by McFarland [3] his lower bound of (q + 1)/2 may be increased to q - 1. This is a consequence of the fact that for all t, 3 d t d q + 1, there is a configuration of q + 1 mutually intersecting lines with maximum multiplicity t (i.e., some point is incident to t lines and no point is incident to more than t lines). Indeed, if aI, a,, a3,...,a,-, are any t - 1 distinct elements of L = GF(q), then the q + 1 lines .Y= 0, r-1 ,V=WZX+ n (??I-Ui), i=l
m EL,
NONCYCLIC DIFFERENCE SETS
constitute such a configuration.
21
For the point (0,O) lies on the t lines x = 0,
y = six,
l
while for any other point (a, b) there are at most t - 1 values of m in L which satisfy the degree t - 1 polynomial equation f-1
b=ma+
fl
(m-a;).
i=l
Hence no point lies on more than
t
of these q + 1 lines. We thus have the
Remark. McFarland’s construction which in Theorem 5.1 interprets E as L*, L = GF(q), and which employs the spread {Hi} given by the ldimensional L-linear subspaces of E produces at least q - 1 pairwise inequivalent difference sets in any group E x K, where K is an arbitrary group of order q + 2. Of course the difference sets constructed above to produce the stated lower bounds have had corresponding geometric configurations with distinct maximum multiplicities. In general for a given maximum multiplicity there may be many inequivalent difference sets which have that particular maximum. A more realistic lower bound for the number of inequivalent difference sets produced from a given spread would be the number of distinct multisets of multiplicities arising from the configurations of q + 1 mutually intersecting lines in the corresponding translation plane. We have not investigated this question even for the familiar case of the Desarguesian plane.
REFERENCES 1. W. BURNSIDE,“Theory of Groups of Finite Order,” 2nd ed., reprint, Dover, New York, 1955. 2. H. B. MANN, “Addition Theorems,” Interscience, New York, 1965. 3. R. L. MCFARLAND, A family of difference sets in non-cyclic groups, J. Combin. Theorem Ser. A 15 (1973), l-10. 4. P. KESAVA MENON, On difference sets whose parameters satisfy a certain relation, Proc. Amer. Marh. Sot. 13 (1962), 739-745. 5. T. G. OSTROM, “Finite Translation Planes,” Lecture Notes in Mathematics Vol. 158, Springer-Verlag, New York, 1970. 6. R. J. TURYN, Character sums and difference sets, Pacific J. Math 15 (1965), 319-346.
OF COMBINATORIAL
THEORY,
Series A 40, 9-21
(1985)
Variations on a Scheme of McFarland for Noncyclic Difference J. F.
Fort
DILLON
Department of Defense, George G. Meade. Maryland
Communicated
by William
Received
1.
Sets
October
20755
M. Kantor
30, 1980
INTRODUCTION
The k-subset D of the group G of order u is called a (u, k, A, n)-difference element of G has exactly 1” representations as a difference of two elements of D; the parameter n is defined to be equal to k - I.. Using multiplicative notation for the group operation, we have that D is a difference set precisely when it satisfies in the group ring Z[G] the equation
set if every nonidentity
DD'~-"=n l,+%G,
(1.1)
where D'- ‘) denotes the set of the inverses of elements of D and 1, denotes the identity element of G. McFarland [3] has given a construction for difference sets having the parameters
(1.2)
where q is any prime power and s is any positive integer. These difference sets are in the groups of the form
G=ExK, 9 0097-3165185
$3.00
Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
10
J. F. DILLON
where E is the elementary abelian group of order qs+’ and K is any group of order r+l, r=(q’+’ - 1)/( q - 1). In particular, this construction produces Hadamard difference sets in any group of order 22(s+‘) which contains Z;+ ’ as a direct factor. In this brief paper we point out two variations on McFarland’s scheme; the first extends the construction to a much larger class of groups and the second produces many more inequivalent difference sets in some of the same groups. In Section 2 we show that the elementary group E need not be a direct factor of G; indeed, essentially the same construction produces difference sets with parameters (1.2) in any group G of order u whose center contains E. In particular, G may be any abelian group of order u which contains E. Taking q = 2, we thus obtain Hadamard difference sets in all groups z; x zp+2.
These difference sets (in abelian groups of order 22s+2 and exponent 2s+2) demonstrate the sharpness of the exponent bound established by Turyn [6] in 1965. Turyn himself gave examples of such difference sets in the case s = 1; our construction produces the first known examples of such difference sets for all s > 1. We present these and other general results for 2groups in Section 3. In Section 4 we restrict our attention to groups of order 16 (the case s = 1) showing that the basic construction may be extended to groups which contain E = Z, x Z, as a normal subgroup. This general construction produces difference sets in ten of the fourteen groups of order 16. We complete this section by showing that the cyclic and dihedral groups are the only groups of order 16 which do not contain a nontrivial difference set. The last section presents our second variation of the basic construction which applies when s = 1 and q is not a prime. Whereas McFarland’s construction employs the l-dimensional subspaces of E regarded as a 2-dimensional GF(q)-linear space, we observe that any spread (i.e., family of q + 1 subgroups of order q the intersection of any two of which is the identity) for E will do as well. Furthermore, in the groups G = E x K, q odd, difference sets produced from inequivalent spreads are themselves inequivalent so that McFarland’s lower bound of (q + 1)/2 for the number of inequivalent difference sets in G may be multiplied by the number of inequivalent spreads for E. We also point out that in the Desarguesian case considered by McFarland his lower bound may be improved to q - 1 and that even this is a gross underestimate for most q. The details of the more general construction are given in the next section. We employ, for the most part, the terminology and notation of McFarland [3].
NONCYCLIC
DIFFERENCE
11
SETS
2. THE CONSTRUCTION Let q be a prime power and let s be a positive integer. Let E be the elementary abelian group of order qsfl which we regard as a vector space of dimension s + 1 over the finite field F= GF(q). Let H,, HI,..., H,, r = (4‘+ ’ - l)/(q - l), be the hyperplanes (i.e., s-dimensional subspaces) of E. Since every nonidentity element of E is contained in precisely (q’ - l)/ (q - 1) hyperplanes we have in the group ring Z[E],
i;, H,=rl.+
{(q”- l)/(q-
1)) (E- 1~).
(2.1)
Let G be any group of order u = qs+ ‘(r + 1) which contains E as a normal subgroup and let G=g,E+g,E+
... +g,+lE
be the coset decomposition of G with respect to E; i.e., {g,, gz,..., gl+, } is any set of representatives of the cosets of E in G. If we regard the cosets g,E as being the elements of the factor group G/E, then the elements
g,E, g,&., g,E constitute a (trivial) (r + 1, r, r - 1, 1)-difference set in G/E. Equivalently, in the group ring Z[G] we have i g,g;‘E=E+(r-l)G=rE+(r-l)(G-E). i.i= 1
(2.2)
Now let D be the subset of G given by D=g,H,+g,H,+
... +g,H,,
so that DD’-“=
i i,j=
g,H,H,g;’ 1
=q’ i
giHig,:‘+q”-’
i=l
i i,j=
i #j
g,g;‘E. 1
By (2.2) the second of these sums is (r - l)(G - E). Thus, D is a difference set in G if and only if qs 1 giHig;’ ,=I
=q”rl.+q”-‘(r-
1) E*,
12
J. F. DILLON
where E* denotes the set of nonidentity elements q(q” - l)/(q - l), this condition is equivalent to ;$I g,H,g;’
=rl,+s
of E. Since r - 1 =
E*;
i.e., every nonidentity element of E must belong to exactly (q’ - l)/(q - 1) of the subgroups gjHig; ‘. Of course, this condition is guaranteed (indeed reduces to (2.1)) if the map Hi --+g, H,g; ’
happens to be a permutation foregoing in the
of the hyperplanes of E. We summarize
the
THEOREM. Let q be a prime power and let s be any positive integer. Let E be the elementary abelian group of order qS+I (which we regard as a vector space of dimension s + 1 over GF(q)), and let H, , Hz,..., H,, r= (q’+’ - l)/(q - l), be the s-dimensional subspacesof E. Let G be a group of order q”+ ‘(r + 1) which contains E as a normal subgroup and let g,, g2,..., g, be elements of G lying in distinct cosets of E. Let D=g,H, +g,H,+ “’ + g,H,. If the coset representatives gi should have the property that { giHTg,: I : 1 < i < r} is a l-design on the point set E*, then D is a difference set with parameters (1.2). COROLLARY. If the subgroup E of G lies in the center of G, then D is a difference set for all choices of the coset representatives g,, g2,..., g,.
McFarland’s result in which E is assumed to be a direct factor of G is, of course, a special case of this corollary. We observe that when q is a prime the family (Hj} of hyperplanes of E is invariant under conjugation by any element of G and if g,, gZ,..., g, are elements of G with no two in the same coset of E then the r x r array A = W$g,?),
1 Q i,j< r,
has the property that every row is a permutation then clear.
of {Hi}.
The following is
COROLLARY. The group G contains a nontrivial difference set if the corresponding array A has a transversal; i.e., there exists a permutation z of { 1, 2, 3,..., r} such that the r entries {A(i, z(i)): 1 < i < r> are distinct.
We shall have occasion to use this last result in Section 4 where we determine the groups of order 16 which contain difference sets.
13
NONCYCLIC DIFFERENCE SETS 3. DIFFERENCE SETS IN ~-GROUPS
The results of the previous section are particularly special case q = 2 when the parameters become
interesting
(u, k, A, n) = (z2”+ 2, z2’+ 1- 2”, z2” - 2”, z2”).
in the (3.1)
Mann has shown [2] that any nontrivial difference set in a a-group (indeed, any SBIBD having u a power of 2) must (up to complementation) have parameters of the form (3.1). Since these parameters satisfy the relation v=4n.
any difference set in a 2-group is necessarily a Hadamard difference called because the incidence matrix of the associated block design, O’s and l’s replaced by l’s and - l’s, respectively, is a Hadamard Indeed, if we associate with any element D in Z[G] the D* = G - 20, we see that the equation DD”‘=nl
is equivalent
G
set-so with its matrix. element
+r”G *
to D*D*’
“=4nl,
+ (v-412) G.
which implies the previous remark. Thus, in a 2-group G of order v = 22sc2 a subset D is a difference set precisely when its associate D* satisfies D*D*‘-‘I=
ICI 1,.
Let &’ denote the class of all finite groups which contain a Hadamard difference set. X contains the group of order 1 and both groups of order 4 the difference sets in which are singletons and their complements. It is well known that Y is closed under the taking of direct products; indeed, if G is the (not necessarily direct) product of two “disjoint” subgroups G, and G, containing Hadamard difference sets D, and D, respectively, then D = D, (G, - D,) + (G, - 0,) D, has associate D* = DTD,* which satisfies (DTDT)(D:‘-“Dr~“)=DTIGzI
lczDT’-“=
IG, I l,, lG21 lGz= ICI 1,.
We thus have the Remark. The class X’ of groups which contain a Hadamard difference set is closed under arbitrary products; i.e., if G = G, G, with G, and G, in 2 and IG, n G, I = 1, then G also belongs to &‘. Since X contains the groups of order 4 we have the
14
J. F. DILLON
COROLLARY. (Turyn and Kesava Menon). is a direct product of groups of order 4.
In his remarkable
2 contains any group which
paper [6] of 1965 Turyn established the
THEOREM. If the abelian group G of order 22S+2 belongs to &‘, then G has exponent at most 2” + ‘.
Turyn himself [6] gave examples of such difference sets in the group Z2 x Z, (i.e., the case s = 1) and asked (private communication) if his exponent bound was sharp for larger s. McFarland’s construction produces difference sets in Z2+ l x Z=>+I which has exponent 2Sf ‘, within a factor of 2 of Turyn’s bound. Our construction demonstrates the sharpness of Turyn’s bound for all s. The theorem of the previous section takes the following simpler form in the case q = 2. THEOREM. Let E be the elementary abelian group of order 2” + I and let H, , H2 ,..., H,, r = 2”+ ’ - 1, be the subgroups of E of order 2”. Let G be a group of order 22sf2 which contains E as a normal subgroup and let g,, g2,..., g, be elements of G lying in distinct cosets of E. Let D =g, H, + g,Hz + . . . +g, H,. If the coset representatives gi should have the property that {g,H,g;‘:
1 Qi
{Hi: 1
then D is a difference set in G with parameters (3.1). COROLLARY. D is a difference set if E lies in the center of G.
The sharpness of Turyn’s bound is now established by this last corollary which implies the COROLLARY. D is a difference set in the group G=Z;xZ,,+s.
Notice that this group G contains a unique subgroup isomorphic to Z;+ ’ which we take as E. Actually, the factor Z2$+2 of G in this last corollary may be replaced by an arbitrary group of order 2’+=. More generally we have the COROLLARY. Zf the group G of order 22S+2is the direct product of s + 1 nontrivial subgroups, then G contains a Hadamard difference set.
15
NONCYCLICDIFF'ERENCESETS
wherelGil>2foreachi, ldi
which we may take as E in the first corollary to the above theorem.
Q.E.D.
We thus obtain Hadamard difference sets in many different groups of order 22S+2 including many nonabelian groups. We illustrate the construction for the special case
where D2’+2 denotes the dihedral group generated by elements a and b satisfying a2 = 1 = b2’+ ‘, Then 8 = b2’ is an involution the subgroup
of order 2Si2. Let DZE+z be
aba=b-‘.
in the center of DzJ+l and we may take as E
whose cosets in G are represented by the 2’+ ’ elements { 6’: 0 < i < 2”) u {ab’:O
D=
c i=
b’e,H,+ I
23+1-I 1 ab’e,H, I=
2s
is a difference set in G.
4. HADAMARD GROUPSOF ORDER 16
The fourteen groups of order 16 are catalogued in Burnside [l] which also conveniently lists for each group its center. Exactly eight of these groups (including, of course , the four noncyclic abelian groups) have center containing Z2 x Z, which we may take as E to construct difference sets according to our theorem. Of the remaining six groups, exactly two contain a normal Z2 x Z2 which we may take as E to construct difference sets according to our theorem and the following Remark. 582a/40/1-2
Let
G
be
a
group
of
order
16
containing
E=
16
J. F. DILLON
h, > = Z, x Z, as a normal subgroup, and let g,, g,, g, be {l=h,,h,,h2, elements of G which lie in distinct cosets of E. Then the matrix A = [g,h,g;l] 1 < i,j< 3 has a transversal. ProofI E must intersect the center of G so that for some j, the j,th column of A is constant h,. The 3 x 2 submatrix of A comprised of the remaining two columns must then have rows which are permutations of the remaining two h’s and two of these rows must be the same. Then either diagonal constitutes a transversal of this 2 x 2 submatrix; and, taking hjO from the remaining row, we obtain a transversal for all of A. Q.E.D.
Appropriate modification of the above argument the affirmative the first few cases of the following
allows us to answer in
Query. Let G be a group of order 22S+2 containing a normal subgroup E isomorphic to Z;+ ‘. Let Hi, Hz,..., H,, r = 2’+ i - 1, be the hyperplanes 2,..., g, be elements of G lying in distinct cosets of E. Let ofEandletg,,g Let A = (g,H,g,: I), 1 d i,j < r. Must A have a transversal? The four groups of order 16 which do not contain a normal Z, x Z2 are I. II. III. IV. Q-IpQ=p-‘,
G cyclic; i.e., G = (P), PI6 = 1; G dihedral; i.e., G = (P, Q), P8 = 1 = Q2, QPQ = P-‘; G = (P, Q ), P8 = 1 = Q2, QPQ = P3; G generalized quaternion; i.e., G = (P, Q), P8 = I= Q4, Q2=p4.
The following easily verified result shows that groups III and IV, though not yielding to our general theorem still contain difference sets so belong to 2. Remark. Let G be either of the groups given by III and IV. Let D be the subset given by
D = { 1, P2, Q, QP, QP”, QP”}. Then D is a difference set in G.
The cyclic and dihedral groups of order 16 do not contain (nontrivial) difference sets-the cyclic group by Turyn’s exponent theorem and the dihedral group by the following THEOREM. Let H be an abelian group and let G be the generalized dihedral extension of H; i.e., G = (Q, H), Q2 = 1, QhQ = h-’ Vh E H. If G contains a difference set, then so doesevery abelian group which contains H as a subgroup of index 2.
NONCYCLIC
DIFFERENCE
17
SETS
Proof. We have G = H + QH. Let D be a difference set in G which we may write as D=X+QY,
where X and Y are subsets of H. Then DD’-“=
(X+
QY)(X+
QY)‘-“=
(X+ QY)(X’-“+
Y(-“Q)
=Xx’-“+Qyy(~“Q+Qyx’-‘)+xy’-‘)Q =
(xJ+“+
yy(-“1
+2Qyx’-“.
Since D satisfies an equation DD’--“=nl.+AG,
it follows that X and Y must satisfy XX’-“+
YY’-“=nl,+AH; yX’-‘,=dH 2
(*) .
Now let K be any abelian group containing H with [K : H] = 2 and let 8 be any element of K not contained in H. Then K=H+BH.
We define a subset C of K by C=X+BY. Then cc’-‘)=
(X+ eY)(x+ey)'-" = (x+ BY)(X'-"
+ I+“&‘)
=xx(-l’+eyyC-*)e-f
+fjy~(-l)+xy(Gl~~-l
=Xx’-‘)+
yy’-I)+0
{yx’-‘l+e~2(yx(-l))(-l)j
=xX’-“+
yy(-l)+2eypl)
the last equality being a consequence of
y~'-"=~H=e~z(~H)'-"=e-2(y~'~")'-".
18
J. F. DILLON
But then by (*) C satisfies CC’-“=nl,+AH+2e(~H)
2
=nl,+AH+MH
=nl,+X, so that C is indeed a difference set in K.
Q.E.D.
COROLLARY. Zf the cyclic group ZZm does not contain a (nontrivial) ference set, then neither does the dihedral group of order 2m.
dif”
We have thus established the following THEOREM. Of the fourteen groups of order 16 only the cyclic and dihedral groups fail to belong to 2.
5. ANOTHER VARIATION
When s = 1 McFarland’s scheme produces difference sets in the groups of order q2(q+2) of the form G=ExK,
where E is the elementary abelian group of order q2 (regarded as a 2dimensional GF(q)-linear space) and K is any group of order q + 2. These difference sets are of the form
D=g,H,+g,H,+
... +gy+,ffy+l,
where H,, H, ,..., H,, , are the hyperplanes in E and g,, g, ,..., gy+ , are any elements of G lying in distinct cosets of E. That such a set D is a difference set follows (as in Sect. 2) only from the fact that the His constitute a spread for E. We thus have the THEOREM. 5.1. Let q be a prime power and let G = E x K, where E is the elementary abelian group of order q2 and K is an arbitrary group of order be a spread for E and let g,, g2,..., gq+ , be q+2. Let HI, H, ,..., H,,, elements of G lying in distinct cosets of E. Then D = g, H, + is a difference set in G with parameters g,H,+ *.’ +gy+lHy+l (v, k 4 n) = (q2(q + 21, q(q + 11, 4, q2).
19
NONCYCLIC DIFFERENCE SETS
Now suppose that q is odd so that E and K have relatively prime orders and are both invariant under every automorphism of G. Suppose that
and
are difference sets of the type given by the theorem and suppose that D and i.e.,
D’ are equivalent;
for some automorphism
CI of G and some element g of G. Then
1 g;H; = D’ = gD’“’ = c gg; Hia) and, since u fixes E and K, it follows that the two sets {gi} represent exactly the same cosets of E and the image { Hj*)) of a spread for E which must coincide with the spread {Hi.}. spreads (Hi} and (Hi) equivalent if there is an automorphism takes one spread into the other. We thus have the
and {gg;> {Hi} is itself We call the of E which
THEOREM. 5.2. For q odd, inequivalent spreads give rise to inequivalent difference sets.
McFarland’s argument [3] extends to this more general construction to give a lower bound on the number of inequivalent difference sets produced by any fixed spread (Hi}. Let x denote the projection homomorphism fromG=ExKtoEandlet
be any difference set obtained via our construction from the spread {Hi}. Then for each i, 1 < i < q + 1, n(g,) Hi is a coset of Hi in E and the image under z of D, namely,
n(D)=c dir;)Hi, as an element of the group ring Z[E], has coefficient on any element e E E equal to the number of cosets z(gi) Hi containing e. Note that n(D) = 1 n( 8,) Hi may be interpreted
as a “configuration”
in the plane determined
by the
20
J. F. DILLON
spread (the points of this plane are the elements of E and the lines are the cosets in E of the components Hi of the spread). Thus, the coefficient on an element e E E in rc(D) is the number of lines in the configuration incident to e. Clearly the multiset of multiplicities given by the coefficients in the group ring element rc(D) constitutes an invariant for the class of difference sets
equivalent to D. McFarland employed the less discriminating invariant given by the largest coefficient and constructed (q + 1)/2 inequivalent difference sets
by choosing for each such t coset representatives g(lr), gy),..., g!i, property that
with the
n( gj”) E Hi for exactly q + 1 - t values of i, 1 d i 6 q + 1. In geometric terms this means that the configuration rc(D,) has exactly q + 1 - t lines passing through the origin (i.e., containing the identity element of E) and since this number is more than half the number of lines in the configuration no other point can have a coefficient so large. Thus, for each t, q + 1 - t is the largest coefficient occuring in rc(D,) and so the difference sets (0,: 0 < t < (q - 1)/2} are indeed inequivalent. We reiterate the foregoing as the THEOREM 5.3. Let q be an odd prime power and let G = E x K, where E is the elementary abelian group of order q2 and K is an arbitrary group of order q + 2. Let S(q) be the number of inequivalent spreadsfor E. Then G contains at least S(q) x (q -t 1)/2 pairwise inequivalent difference sets.
For the Desarguesian case assumed by McFarland [3] his lower bound of (q + 1)/2 may be increased to q - 1. This is a consequence of the fact that for all t, 3 d t d q + 1, there is a configuration of q + 1 mutually intersecting lines with maximum multiplicity t (i.e., some point is incident to t lines and no point is incident to more than t lines). Indeed, if aI, a,, a3,...,a,-, are any t - 1 distinct elements of L = GF(q), then the q + 1 lines .Y= 0, r-1 ,V=WZX+ n (??I-Ui), i=l
m EL,
NONCYCLIC DIFFERENCE SETS
constitute such a configuration.
21
For the point (0,O) lies on the t lines x = 0,
y = six,
l
while for any other point (a, b) there are at most t - 1 values of m in L which satisfy the degree t - 1 polynomial equation f-1
b=ma+
fl
(m-a;).
i=l
Hence no point lies on more than
t
of these q + 1 lines. We thus have the
Remark. McFarland’s construction which in Theorem 5.1 interprets E as L*, L = GF(q), and which employs the spread {Hi} given by the ldimensional L-linear subspaces of E produces at least q - 1 pairwise inequivalent difference sets in any group E x K, where K is an arbitrary group of order q + 2. Of course the difference sets constructed above to produce the stated lower bounds have had corresponding geometric configurations with distinct maximum multiplicities. In general for a given maximum multiplicity there may be many inequivalent difference sets which have that particular maximum. A more realistic lower bound for the number of inequivalent difference sets produced from a given spread would be the number of distinct multisets of multiplicities arising from the configurations of q + 1 mutually intersecting lines in the corresponding translation plane. We have not investigated this question even for the familiar case of the Desarguesian plane.
REFERENCES 1. W. BURNSIDE,“Theory of Groups of Finite Order,” 2nd ed., reprint, Dover, New York, 1955. 2. H. B. MANN, “Addition Theorems,” Interscience, New York, 1965. 3. R. L. MCFARLAND, A family of difference sets in non-cyclic groups, J. Combin. Theorem Ser. A 15 (1973), l-10. 4. P. KESAVA MENON, On difference sets whose parameters satisfy a certain relation, Proc. Amer. Marh. Sot. 13 (1962), 739-745. 5. T. G. OSTROM, “Finite Translation Planes,” Lecture Notes in Mathematics Vol. 158, Springer-Verlag, New York, 1970. 6. R. J. TURYN, Character sums and difference sets, Pacific J. Math 15 (1965), 319-346.