SPECIAL SUBSETS OF DIFFERENCE SETS WITH PARTICULAR EMPHASIS ON SKEW HADAMARD DIFFERENCE SETS

SPECIAL SUBSETS OF DIFFERENCE SETS WITH PARTICULAR EMPHASIS ON SKEW HADAMARD DIFFERENCE SETS ROBERT S. COULTER AND TODD GUTEKUNST Abstract. Ths artcle ntroduces a new approach to studyng dfference sets va ther addtve propertes. We ntroduce the concept of specal subsets, whch are nterestng combnatoral objects n ther own rght, but also provde a mechansm for measurng addtve regularty. Skew Hadamard dfference sets are gven specal attenton, and the structure of ther specal subsets leads to several results on multplers, ncludng a categorsaton of the full multpler group of an abelan skew Hadamard dfference set. We also count the number of ways to wrte elements as a product of any number of elements of a skew Hadamard dfference set. 1. Introducton Let G be a group, wrtten multplcatvely. Choose any subset D of G and generate all quotents xy 1 wth x, y D and x y. It s qute lkely that n performng ths task, some elements wll be generated more often than others. However, f every non-dentty element of G s generated some constant number of tmes, then we call D a dfference set. Classcally, D s called a (v, k, λ)-dfference set n G, where v = o(g), k = D and λ s the number of tmes each non-dentty element s generated. It s clear For any nteger n, we may defne the set k(k 1) = λ(v 1). (1) D (n) := {d n : d D}. In partcular, wth n = 1 we obtan the set D ( 1) consstng of the nverses of the elements of D. If D s a dfference set, t s not generally true that D (n) s a dfference set. Let D be a dfference set n G. For any element g G, the set Dg := {dg : d D} s also a dfference set wth the same parameters as D. The sets Dg are called translates of D. A multpler for D s any automorphsm of G that maps D onto one of ts translates. If r s an nteger coprme to o(g) such that the automorphsm φ r : x x r nduced by r on G s a multpler for D, then φ r s a numercal multpler for D. Dfference sets arse naturally n the study of combnatoral desgns. Specfcally, f a symmetrc balanced ncomplete block desgn wth parameters (v, k, λ) admts a regular automorphsm group G, then the ponts of the desgn can be labelled by the elements of G and any block of the desgn forms a (v, k, λ)-dfference set n G. Conversely, any dfference set D gves rse to such a desgn, called the development 1

ROBERT S. COULTER AND TODD GUTEKUNST Des. Codes Crytogr. 53 (009), 1-1 of D. The ponts of the desgn are the elements of G and the blocks of the desgn are the translates of D. Dscusson of combnatoral desgns leads naturally to the concept of equvalence of dfference sets. Two dfference sets D 1 and D n G are equvalent f one can be mapped onto the other by means of a group automorphsm and a translaton: D φ 1 g = D for some φ Aut(G) and g G. Adjectves attached to dfference sets often descrbe ether the group n whch they lve or the desgn they generate. For example, a dfference set D G s called abelan or cyclc f the group G s abelan or cyclc, respectvely. When λ = 1, the development of a (v, k, 1)-dfference set s a projectve plane. Consequently, such dfference sets are called planar. A dfference set s called skew f D D ( 1) =, whle t s called reversble f D = D ( 1). A classcal example of a dfference set s the set {1,, 4} n Z 7, +. Ths set s cyclc, planar and skew. More generally, f q s any prme power congruent to 3 (mod 4), then the non-zero squares of the fnte feld F q form a (q, q 1, q 3 4 )- dfference set n F q, +. Known as the Paley dfference sets, these skew dfference sets are partcularly specal, as they are examples of skew dfference sets wth k as large as possble wth respect to v. A dfference set D n G s called a skew Hadamard dfference set (SHDS) f D D ( 1) {1} = G. For many years, the Paley dfference sets were the only known abelan skew Hadamard dfference sets. However, n 005 Dng and Yuan [4] dscovered a new class and showed them to be dstnct from the Paley class n some small cases. Further examples have snce been found, see Dng, Wang and Xang [3]. It s known a skew Hadamard dfference set can only exst n a group of order q = p m, where p s a prme and q 3 mod 4; see Johnsen [6]. For a recent survey on dfference sets, see Xang [9]. Let D be a subset of G, and let a be an arbtrary element of G. We defne the specal subsets of D wth respect to a by A a,d = {x D : a = xy 1 for some y D}, B a,d = {y D : a = xy 1 for some x D}, C a,d = {x D : a = xy for some y D}. The set D wll usually be clear from context, so we may omt the subscrpt D when we wrte the specal subsets. Intutvely, the cardnaltes of the sets A a and C a measure the number of ways to wrte a as a quotent and product n D, respectvely. In ths context, a set D G s a dfference set f and only f A a s constant for all nondentty elements a G. The paper s set out as follows. In Secton we brefly consder specal subsets for an arbtrary set. We ntroduce the man body of theory concernng specal subsets of skew Hadamard dfference sets n Secton 3. In Secton 4 we present several results on multplers, ncludng a categorsaton of the full multpler group of an abelan SHDS. We also establsh that abelan SHDS satsfy the Multpler Conjecture usng an unsuspected regularty of products wthn abelan SHDS. In Secton 5 we compute the product of all elements of an abelan skew Hadamard dfference set, as well as the products of all elements wthn each specal subset.

SPECIAL SUBSETS OF DIFFERENCE SETS 3 Des. Codes Crytogr. 53 (009), 1-1. Intal observatons on specal subsets Our frst two results concernng specal subsets are for arbtrary subsets of a group G. Both, however, have mplcatons to dfference sets; the frst to skew Hadamard dfference sets, and the second to reversble dfference sets. Lemma.1. Let D be a subset of G. Then A a C a = for all a G f and only f D D ( 1) =. Proof. If x A a C a for some a G, then a = xy 1 and a = xz for some elements y, z D. But then x 1 a D D ( 1). Conversely, f A a C a = for all a G, then n partcular we have A 1 C 1 =. Now A 1 = D and C 1 D, so C 1 =. Ths mples D s skew. Lemma.. Let D be a subset of G. Then A a = C a for all a G f and only f D = D ( 1). Proof. Suppose D = D ( 1). If a = xy 1 s a representaton for a as a quotent n D, then a = x(y 1 ) s also a representaton for a as a product n D, snce y 1 D by hypothess. Hence A a = C a for any a G. Now suppose A a = C a for all a G. In partcular, D = A 1 = C 1, whch s equvalent to D = D ( 1). We now consder how the values A a and C a are affected when we translate the set D to the set Dg. If a = xy 1 s a representaton for the element a as a quotent n D, then clearly a = (xg)(yg) 1 s a representaton for a as a quotent n Dg. Conversely, any representaton for a as a quotent n Dg corresponds to a unque representaton for a as a quotent n D. Equvalently, the numbers A a are nvarant under translaton. But what about the numbers C a? If G s nonabelan, then lttle can be sad. But f G s abelan, then whenever a = xy s a representaton for a as a product n D, (xg)(yg) s a representaton for ag as a product n Dg. Conversely, any representaton for ag as a product n Dg corresponds to a unque representaton for a as a product n D. Thus C a,d = C ag,dg. So whle the numbers C a are n general not nvarant under translaton, t s clear that the multset {{ C a : a G}} s nvarant under translaton. Now, f D s a subset of an abelan group G, what are the possble values C a,d? Agan f G s nonabelan, then lttle can be sad. If G s abelan, then we have the followng theorem. Theorem.3. For any subset D of an abelan group G, { (Dg) (Dg) ( 1) : g G} { C a,d : a G}. If o(g) s odd, then these sets are necessarly equal. Proof. Let g G and suppose (Dg) (Dg) ( 1) = t for some nteger t. Then C 1,Dg = t. Whenever we can wrte 1 = (d 1 g)(d g) as a product n Dg, then g = d 1 d and conversely. Hence C g,d = t. In partcular, when o(g) s odd, then {g : g G} = G, so every value C a,d s also a value of (Dg) (Dg) ( 1) for some g G.

4 ROBERT S. COULTER AND TODD GUTEKUNST Des. Codes Crytogr. 53 (009), 1-1 We end ths secton wth one fnal lemma concernng specal subsets of a skew set D. Clearly, for D G to be skew, D cannot contan any pars of nverses nor any element whch s ts own nverse. If D s a skew set that s not properly contaned n any other skew set n G, then we call D a maxmal skew set. If v = o(g) s odd, any maxmal skew set has sze v 1. Lemma.4. Let G be a group of odd order v and let D be a maxmal skew subset of G. Then { v 3 A a + C a = f a D, v 1 f a / D. Proof. Snce o(g) s odd, every nondentty element a of G has an nverse dstnct from tself, so precsely one of a, a 1 s n D. Let a D. For each x D, there s a unque y G such that a = xy. If y D, then x C a. If y / D and x a, then y 1 D, so x A a. However, when x = a we have y = 1, and 1 / D. Hence A a + C a = D 1 = v 3. Now let a / D. Ether a = 1 or a D ( 1). If a = 1, then A a = D whle C a =, so A a + C a = v 1. If a D( 1), then by the same argument used above, every x D s ether n A a or C a. Hence A a + C a = v 1. 3. Specal subsets of skew Hadamard dfference sets We now turn to skew Hadamard dfference sets. Specfcally, we outlne the fundamental role the specal subsets play n ther constructon and behavor. The frst step n ths approach s to recognse we may count the number of ways to generate each group element as a product wthn a skew Hadamard dfference set. Theorem 3.1. If D s a (v, k, λ) skew Hadamard dfference set, then every element of D can be wrtten n precsely λ ways as a product n D, whle every element of D ( 1) can be wrtten n λ + 1 ways as a product n D. Proof. If D s a (v, k, λ) skew Hadamard dfference set, then (v, k, λ) = (v, v 1, v 3 4 ). Snce D s skew and D = v 1, D s a maxmally skew set n G. Hence we may apply Lemma.4. Note A a = v 3 4 for all a 1. If a D, then whle f a D ( 1), then C a = v 3 C a = v 1 v 3 4 v 3 4 = λ, = λ + 1. Corollary 3.. If D s an abelan skew Hadamard dfference set, then every translate Dg of D satsfes (Dg) (Dg) ( 1) {λ, λ + 1}. Proof. By Theorems.3 and 3.1, the quantty (Dg) (Dg) ( 1) may take one of only three values: 0, λ, or λ + 1. However, f (Dg) (Dg) ( 1) = 0 then g cannot be wrtten as a product n D. If g 1, then ths s mpossble, and thus any nontrval translate Dg of D satsfes (Dg) (Dg) ( 1) {λ, λ + 1}.

SPECIAL SUBSETS OF DIFFERENCE SETS 5 Des. Codes Crytogr. 53 (009), 1-1 If D s a (v, k, λ) skew Hadamard dfference set, and f a D, then A a = C a = λ = k 1. Note a s not n ether A a nor C a, snce 1 / D. By Lemma.1 the sets A a and C a are dsjont, so by sze consderatons D must be the dsjont unon of A a, C a, and {a}. Smlarly, f a D ( 1), then D s the dsjont unon of A a and C a. In other words, any non-dentty element a G nduces a partton of D by the specal subsets A a and C a (and possbly the sngleton {a}). We now show that among dfference sets, ths property s characterstc only of skew Hadamard dfference sets. Theorem 3.3. A (v, k, λ) dfference set D s skew Hadamard f and only f the followng condtons hold: () For any a D, D s the dsjont unon of A a, C a, and {a}. () For any a / D, D s the dsjont unon of A a and C a. Proof. A combnaton of Lemma.1 and Theorem 3.1 proves any skew Hadamard dfference set satsfes the two condtons. Conversely, any dfference set D satsfyng these two condtons must be skew by Lemma.1, so C 1 =. Our task s to show k = v 1, as all else wll then follow from Theorem 3.1. We have C a = C 1 + C a + C a a G a D a/ (D {1}) = 0 + k(k λ 1) + (v k 1)(k λ) = k(v ) λ(v 1) = k(v ) k(k 1) = k(v k 1). But for any x D, x C xy for each y D. Lettng x vary over D, we have C a = k. a G Hence k = v 1, whch completes the proof. Whle the sets B a appear to play less of a role n ths approach, they do exhbt a predctable regularty n the abelan case: they splt as evenly as possble between the sets A a and C a. Theorem 3.4. Let D be an abelan skew Hadamard dfference set n G, and let a G, a 1. If λ s even, then A a B a = B a C a = λ. If λ s odd and a D, then A a B a = B a C a = λ 1. If λ s odd and a / D, then A a B a = λ 1 and B a C a = λ+1. Proof. Fx a 1. Select any x A a such that a = xy 1 for some y B a. For every x A a wth x 1 x satsfyng a = x y 1 for some y B a, we have (xx 1 )(y y 1 ) = 1. Snce D s skew Hadamard, precsely one of xx 1 or y y 1 must be n D. In the former case xx 1 D and y y 1 D ( 1), so x C x and y C y. So, n ths case, we fnd In the latter case xx 1 conclude A a C x = B a C y. D ( 1) and y y 1 D, so x A x and y A y. Here we A a A x = B a A y.

6 ROBERT S. COULTER AND TODD GUTEKUNST Des. Codes Crytogr. 53 (009), 1-1 The conclusons of the two cases are equvalent by Lemma.1. Usng a = a a 1, we therefore have A a C a = B a C a for any a 1. Select any par x, y C a satsfyng a = xy. For any par x, y C a wth x x satsfyng a = x y we have (xx 1 we fnd Settng x = y = a shows Now )(yy 1 ) = 1. Usng a smlar argument as above C a C x = C a A y. C a C a = C a A a = B a C a. () C a = { λ f a D, λ + 1 f a D ( 1). The partton {A a, C a } of D and () shows { C a C a + 1 f a D, C a = C a C a f a D ( 1). Solvng for B a C a = C a C a va (3) yelds the correspondng clams. 4. Multpler results for abelan skew Hadamard dfference sets We now consder how the addtve structure of skew Hadamard dfference sets descrbed n Secton 3 affects the multpler groups for abelan skew Hadamard dfference sets. Throughout ths secton p s a prme, p 3 mod 4, and D s an abelan skew Hadamard dfference set n G, wth exp(g) = p s. The strongest result known concernng the multpler group of D was establshed by Camon and Mann [1], who showed the quadratc resdues modulo p s, Q p s, are precsely the numercal multplers of D. Our frst result on multplers s a weaker, though often equvalent, verson of Camon and Mann s result. Theorem 4.1. Let D be an abelan (v, k, λ) skew Hadamard dfference set n G. If v 3 mod 8, then D () = D ( 1), and hence D (4) = D. If v 7 mod 8, then D () = D. Proof. Let a D. If a = xy, then a = yx snce G s abelan, so the ways to wrte a D as a product of two dstnct elements of D come n pars. If v 3 mod 8, then λ = v 3 4 s even. It follows from Theorem 3.1 that a x for any x D, or equvalently, D D () =. Hence D () = D ( 1), as clamed. If v 7 mod 8, then λ s odd, so there must exst some x D such that a = x. Hence D () = D. It s often true that for prmes p 3 mod 8, 4 s a multplcatve generator for all quadratc resdues modulo p m. Lkewse, for prmes p 7 mod 8, s often a multplcatve generator of the quadratc resdues modulo p m. For any such prme, Theorem 4.1 s equvalent to the stronger result of Camon and Mann. However, there are prmes congruent to 3 mod 4, the smallest of whch s 43, for whch nether nor 4 generate all quadratc resdues. Thus, ths result s clearly weaker. We note, however, the fundamental nature of the proof, how t follows mmedately from Theorem 3.1, and how t does not rely on the assumpton that G s a p-group. (3)

SPECIAL SUBSETS OF DIFFERENCE SETS 7 Des. Codes Crytogr. 53 (009), 1-1 In ths sense, one can vew our proof as provdng a fundamental reason as to why the quadratc resdues must be multplers of a skew Hadamard dfference set. As the quadratc resdues modulo p s comprse the numercal multpler group of D, they form a subgroup of the full multpler group of D. Our next result s a categorsaton of the full multpler group of an abelan skew Hadamard dfference set. Theorem 4.. Let D be an abelan skew Hadamard dfference set n a group G of exponent p s. The full multpler group of D s precsely the stablzer of D n Aut(G). Furthermore, the quadratc resdues modulo p s form a subgroup Q p s of the stablzer of D. Consequently, D s the unon of orbts of the acton of Q p s on G. Proof. By Corollary 3., no translate of D s skew. As a multpler for D s frstly an automorphsm, t must map skew sets to skew sets. Consequently, any multpler of D must n fact fx D and any automorphsm whch does so s clearly a multpler. The remanng clams are clear. Our fnal multpler result, Theorem 4.5, wll follow from the observaton that one may easly count the number of ways to wrte elements as the product of any number of elements of a skew Hadamard dfference set D. The computatons are most easly carred out n the group rng ZG. Theorem 4.3. Let D G be a (v, k, λ) skew Hadamard dfference set. Set t = (λ + 1) + λ(λ + 1)G. Then for any nteger j, we have D j = a j D + (λ + 1)a j 1 D ( 1) + a j t, (4) where the sequence {a } satsfes a 0 = 0, a 1 = 1, a = λ and a = λa 1 +ta 3 for 3. Proof. We nduct on j. When j = we have D = λd + (λ + 1)D ( 1), whch s correct by Theorem 3.1. Now suppose j and the formula (4) s correct for j. Then D j+1 = DD j = D(a j D + (λ + 1)a j 1 D ( 1) + a j t) Ths completes the nducton. = a j D + (λ + 1)a j 1 DD ( 1) + a j td = a j (λd + (λ + 1)D ( 1) ) + (λ + 1)a j 1 (λg + (λ + 1)) + a j td = (λa j + ta j )D + (λ + 1)a j D ( 1) + a j 1 [λ(λ + 1)G + (λ + 1) ] = a j+1 D + (λ + 1)a j D ( 1) + a j 1 t. Introducng the sequence {a j } of coeffcents n Theorem 4.3 allows for relatvely uncomplcated expressons for hgher powers of a skew Hadamard dfference set D. The next lemma provdes a closed formula for these coeffcents. Lemma 4.4. The sequence {a j } defned recursvely by a 0 = 0, a 1 = 1, a = λ, and a j = λa j 1 + ta j 3 for 3 satsfes a j = ( ) j 1 λ j 3 1 t. (5) 0

8 ROBERT S. COULTER AND TODD GUTEKUNST Proof. Base cases j = 0, 1, are easly confrmed. Suppose the formula (5) holds for all ntegers 0 j l for some nteger l. We must prove the formula holds for j = l + 1. We have Des. Codes Crytogr. 53 (009), 1-1 a l+1 = λa l + ta l = λ ( ) l 1 λ l 3 1 t + t ( ) l 3 λ l 3 3 t 0 0 = ( ) l 1 λ l 3 t + ( ) l 3 λ l 3 3 t +1 0 0 = ( ) l 1 λ l 3 t + ( ) l m 1 λ l 3m t m, m 1 0 m 1 where we have set m = +1 n the second sum. Note that when m = 0, ( ) l m 1 m 1 = 0, so we may let the second sum run over m 0 rather than m 1. Relabellng m as n the second sum, we have a l+1 = ( ) l 1 λ l 3 t + ( ) l 1 λ l 3 t 1 0 0 = [( ) ( )] l 1 l 1 + λ l 3 t 1 0 = ( ) l λ l 3 t, 0 whch agrees wth the formula. Ths completes the nducton. Theorem 4.5. Let D G be an abelan (v, k, λ) skew Hadamard dfference set. Then any odd prme dvsor of λ + 1 s a multpler of D. Proof. Suppose q s an odd prme dvsor of λ + 1. Then t 0 mod q, and hence from Lemma 4.4 we see a j λ j 1 mod q. We thus obtan D q = a q D + (λ + 1)a q 1 D ( 1) + a q t λ q 1 D mod q D mod q. Snce D p D (p) mod p for any prme p (see [7] Lemma 3.3), t follows D (q) D mod q. Ths s possble only f D (q) = D. Hence q s a multpler for D. Theorem 4.5 s connected to one of the well-known problems of multpler theory. The parameters of a skew Hadamard dfference set satsfy k = λ+1, so λ+1 = k λ. For any (v, k, λ) dfference set, the quantty k λ s typcally called the order of the dfference set. A famous theorem of Hall [5] on cyclc groups, generalzed to abelan groups by Chowla and Ryser [], states f D s an abelan dfference set and q > λ s a prme dvdng k λ but not v, then q s a multpler of D. It s generally beleved, however, that the hypothess q > λ s unnecessary. The Multpler Conjecture therefore states that any prme dvsor of k λ s a multpler for the abelan (v, k, λ) dfference set D. Theorem 4.5 shows ths conjecture s true for skew Hadamard dfference sets. Ths was shown by Xang n [8, Corollary..6].

SPECIAL SUBSETS OF DIFFERENCE SETS 9 5. Products of elements wthn specal subsets For any subset D of a group G, we may defne a drected graph Γ D as follows: the elements of D are the vertces, and the drected edge (x, y) s n Γ D f and only f xy 1 D. Lemma 5.1. If D s an abelan skew Hadamard dfference set, then the drected graph Γ D s a λ-regular tournament. Des. Codes Crytogr. 53 (009), 1-1 Proof. For dstnct elements x, y of G, xy 1 D f and only f yx 1 / D, snce D s skew Hadamard. It follows that Γ D s a tournament. Let x D, and suppose (x, y) s an edge of Γ D. Then xy 1 = d for some d D, hence y C x. Conversely, any y C x corresponds to an edge (x, y) of Γ D, so the outdegree of vertex x s C x, whch equals λ. As x was arbtrary, we see Γ D s a λ-regular tournament. The graph Γ D provdes a useful mechansm for provng results about certan large products wthn D, whch we now descrbe. As an edge (x, y) n Γ D s naturally assocated to the element xy 1, we can assocate the walk x y z n Γ D to the product (xy 1 )(yz 1 ) = xz 1. In ths fashon we may ascrbe a value vw 1 to a walk of any length from the vertex v to the vertex w. Clearly the value of a walk from v to w s n D f and only f (v, w) s an edge of Γ D. Also clear s the fact that any crcut has value 1. Any regular tournament s necessarly Euleran, so let C be an Euler crcut n Γ D. Now C has value 1, but as we traverse C each element of D s generated precsely λ tmes. As G s abelan, we have ( λ d λ = d) = 1. But then d D d D ( gcd (v,λ) d) = 1. d D Recall that G s a p-group for some prme p 3 mod 4. It s easy to show { 3 f p = 3, gcd (v, λ) = 1 t p 3. We have proven the followng: Lemma 5.. If D G s an abelan skew Hadamard dfference set wth 3 o(g), then d = 1. d D Under the hypotheses of Lemma 5., we may compute the product of all elements n each of the specal subsets of D. Let a D and suppose λ s even. Then D () = D ( 1), so there s no x D satsfyng x = a. It follows z = a λ/ (6) z C a

10 ROBERT S. COULTER AND TODD GUTEKUNST Des. Codes Crytogr. 53 (009), 1-1 By Lemma 5. we have ( d = a d D z C a z )( x A a x ) = 1, hence x = a λ 1. (7) x A a It then follows y = a λ λ 1. (8) y B a If λ s odd, then there exsts b D such that b = a. In ths case, smlar arguments yeld z = a λ 1 b, (9) z C a x = a λ 1 1 b 1, (10) x A a y = a λ λ 1 1 b 1. (11) y B a One may easly derve smlar expressons for specal subsets wth respect to an element of D ( 1). In Secton 3 we observed any non-dentty element a nduces a partton of a skew Hadamard dfference set D nto ts specal subsets A a and C a (and possbly the sngleton {a}). Usng the above products, we may now show that n the abelan case, no two dstnct elements may nduce the same partton. Theorem 5.3. If D G s an abelan skew Hadamard dfference set, then no two dstnct elements of G nduce the same partton of D nto specal subsets. Proof. If a, b D, then the parttons wth respect to a and b are the same f and only f a = b, as the sngletons {a} and {b} are cells of ther respectve parttons. Smlarly, f a D and b D ( 1), then ther respectve parttons are clearly not the same. So suppose a 1 and b 1 are elements of D ( 1) nducng the same partton of D. Then A a 1 = A b 1, whch means B a = B b. If λ s even, then a 3 λ 1 = y = y = b 3 λ 1, y B a y B b hence a 3 λ+1 = b 3 λ+1. Note o(g) = p m for some prme p, so f we can show gcd ( 3 λ + 1, p) = 1, then we may conclude a = b. To the contrary, suppose gcd( 3 λ + 1, p) > 1, or equvalently suppose p dvdes 3 λ + 1. Wth λ = pm 3 4 we have 3 8 (pm 3) + 1 = tp for some nteger t. Thus 3p m 8tp = 1, so p 1, a contradcton. Hence gcd( 3 λ + 1, p) = 1 and we conclude a = b. Now suppose λ s odd. Then B a = B b mples a λ+1 = b λ+1. As before, t suffces to show gcd( λ+1, p) = 1. If gcd(λ+1, p) > 1 then p (λ + 1), hence pm + 1 = tp for some nteger t. It follows that p 1, a contradcton. Hence gcd ( λ+1, p) = 1, so a = b.

SPECIAL SUBSETS OF DIFFERENCE SETS 11 Des. Codes Crytogr. 53 (009), 1-1 We conclude wth a strengthenng of Lemma 5.. Suppose D s an abelan skew Hadamard dfference set n the p-group G. Denote by D p the subset of all elements of D of order p. Theorem 5.4. Let G be an abelan group of order p m, exponent p s, admttng a skew Hadamard dfference set D. If p 3 then d = d = 1. d D p s d D\D p s Proof. If p 3, then by Lemma 5. we know d = d Set d D and note o(g) p s 1. We have d D\D p s Say D p s = {d 1,...,d t }. Then gd 1 o(gd 1 ) = p s, we have o(d 1 d t d 1 Suppose g 1. Then d 1 d t d 1 then d 1 d t d 1 d D\D p s g = d D p s d = g 1, d D p s d. d = 1. = d 1 d d t d 1 ) = p s for all = 1,...,t. for any {1,...,t}. Snce s ether n D p s or D ( 1) p. If d s 1 d t d 1 D p s, = d j for some j. Then d d j = d 1 d t = g, so d, d j C g. Note at most one d can satsfy d = g, so the number of values such that d 1 d t d 1 can be no greater than λ+1 = pm +1 8. It s easy to show D p s = t pm p m 1 D p s. As there are t products of the form d 1 d t d 1 and at most pm +1 8 of them are n D p s, there are at least p m p m 1 pm + 1 8 = 3pm 4p m 1 1 8 such products n D ( 1) p. Now f d s 1 d t d 1 = d 1 j, then g = d d 1 j. Snce we assume g 1, we have j and hence d j B g. Thus whch mples A g = pm 3 4 3pm 4p m 1 1, 8 4 p + 5 p m 1. As p > 3, ths s mpossble. Hence g = 1, whch proves the theorem. References [1] P. Camon and H. B. Mann. Antsymmetrc dfference sets. J. Number Theory, 4:66 68, 197. [] S. Chowla and H. J. Ryser. Combnatoral problems. Canad. J. Math., :93 99, 1950. [3] C. Dng, Z. Wang, and Q. Xang. Skew Hadamard dfference sets from the Ree-Tts slce symplectc spreads n PG(3, 3 h+1 ). J. Combn. Theory Ser. A, 114:867 887, 007. [4] C. Dng and J. Yuan. A famly of skew Hadamard dfference sets. J. Combn. Theory Ser. A, 113:156 1535, 006.

1 ROBERT S. COULTER AND TODD GUTEKUNST [5] M. Hall, Jr. Cyclc projectve planes. Duke Math. J., 14:1079 1090, 1947. [6] E. C. Johnsen. Skew-Hadamard abelan group dfference sets. J. Algebra, 4:388 40, 1966. [7] D. Jungnckel. Dfference sets. In Contemporary Desgn Theory, Wley-Intersc. Ser. Dscrete Math. Optm., pages 41 34. Wley, New York, 199. [8] Q. Xang, Dfference Sets: Ther multplers and exstence, Ph.D. thess, Oho State Unversty, 1995. [9] Q. Xang. Recent progress n algebrac desgn theory. Fnte Felds Appl., 11:6 653, 005. Des. Codes Crytogr. 53 (009), 1-1 (Coulter) Department of Mathematcal Scences, Unversty of Delaware, Newark, DE, 19716, Unted States of Amerca. E-mal address: coulter@math.udel.edu (Gutekunst) Department of Mathematcs, Kng s College, Wlkes-Barre, PA, 18711, Unted States of Amerca. E-mal address: toddgutekunst@kngs.edu