The existence of difference sets in abelian 2-groups is a recently settled problem [5]; this note extends the abelian constructs of difference sets to nonabelian groups of order 64

Davis, James A

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University of Richmond

University of RichmondUR Scholarship RepositoryMath and Computer Science Faculty Publications Math and Computer Science1991A Note on Nonabelian (64, 28, 12) Difference SetsJames A. DavisUniversity of Richmond, jdavis@richmond.eduFollow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publicationsPart of the Discrete Mathematics and Combinatorics CommonsThis Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted forinclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information,please contact scholarshiprepository@richmond.edu.Recommended CitationDavis, James A. "A Note on Nonabelian (64, 28, 12) Difference Sets." Ars Combinatoria 32 (1991): 311-14.A Note on Nonabelian (64, 28, 12) Difference Sets J.A. Davis University of Richmond VA 23173 U.S.A. Abstrad. The existence of difference sets in abclian 2-groups is a recently settled problem [5]; this nOIC extends the abelian constructs of difference sets to nonabclian groups or order 64. 1. Introduction. A difference set D in a finite group G (of order v) is a subset of size k so that every nonidentity element of G can be represented .>. times as differences from elements in D. For groups of order a power of 2, the existence of difference sets in the abelian groups is a recenlly settled problem [S]. The nonabelian case is more difficult; this paper discusses a technique for using the scructure of the abelian difference sets to obtain difference sets in nonabelian groups of order 64. It is helpful to consider the ring Z [ G] . If A c G. we will abuse notation by writing A= Eco'EA a' as an element of Z[G]. Also, A<-1> = Eco'EA(a')-1. By the definition of a difference set, D c G is a difference set iff vv<-1> = (k - .>.) 1 + .>.G. In the v = 64 case, this is DDH> = 16( I) + 12G. All groups in this ]Japer will be written multiplicatively, including abelian groups. Subgroups will be denoted (. , . ) , where the generators are included in the brackets. 2. The abelian case. The abelian case Z16 x Z4 is discussed in [l], [2], (4]. Theorem 2.1. Z16 x Z4 has a difference set. We will provide the difference set and leave it to the reader to verify the theorem (either by a straight check or by character theory). All elements of the group can be written as powers of a and b, where a 16 = b4 = I . If we take the subgroup H = (a4 , b2), we can writeG = Ug;H. Take the following subsets of H: Ds The claim is thataD2 Uabfl-J Ua3 D4 UDs Ub.LJS Ua2 D6 Ua6b~ is a difference setinZ16 x Z4. The following lemma is proved in [3]. ARS COMBINATORIA 32(1991), pp. 311 -3 14 Lemma 2.2. Iii;/- j, then D1DJ-1> = 2H. This lemma excludes two important cases, which are covered in the next lemma. Lemma 2.3. (a) Dsn;<-1> = D'sDs1 =4(a1 + b1 ) + 2(a4 + a4 b2 + a11 + a1lb2) (b) D6D~{-l) = fl6D~-t) = 4(a1 + a1 b1 ) + 2(a4 + a"b2 + a12 + a 12b2) (c) DsD;c-tJ + b2 DsD~-1 > = 4H (d) D6D~-1> +a8 b2D6~<-1>=4H. Notice that these lemmas can be considered in the group ring Z[ H] since all the D, are subsets of H. 3. The nonabelian case. Let G be any group of order 64 with a normal subgroup H ~ Z4 x Z2 • We can choose an isomoqilism f : H -+ H. where H is the Z4 x Z2 from the abelian case. Define D; ={/(cl.;) Id; E Di}. Theorem 3.1. A group G with a nonnal subgroup H isomorphic to Z4 x Z2 Jw a difference set if: · (a) there are 4 distinctcosetrepresenmtivesg5 , gfs, g6 , 96 inG/ H so that 9s C§"5)-1 =9s9s 1 (f(b2 )) and 96(96)-1 =9696 1 (f(a1 b2 )), (b) gD;~-l)g-1 = n.D!-l) foreveryg E G, () - D n'(-1)_,_ l - -1-ID n'( -1) ...., n'-=<n -1) __ 1 ...., --1 Dl.,.{D -1) c 9s s s 9s = 9s9s s , 9s s s 9s = 9s9s s s , and the same if we replace 5 by 6. Proof: Pick three distinct coset representatives g2 , g3 , g4 in G / H that are also distinctfromg5 ,'§"s,96.96 · WeclaimthatD = 92D2l.Jg3 D3l.Jg4D4UgsDsU9s n; u 96 D6U g(,D~ is a difference set in G. Consider the group ring equation . D -:::<D -ll ~- D -~D -t>--t (1) : L...J9i I j gj • i,j We can apply Lemmas 2.2 and 2.3 to this situation since f is a group ring isomor-phism from Z [HJ to Z [HJ. If i =I j , Lemma 2.2 implies _D.,.(D -ll--t -c2n)--1 --1(2H) 9; i j 9j =9; 9j =g;gj . Combining Lemma 2.3, (a), and (c), - D -='n <-ll-1-t + _, D'-;:dD -1) --1 9s s s 9s 9s s s 9s - - 1-1v v1<-1> -1--1 v'.,.(D - 11 =g5g5 s s +g5g5 s s - - -1-ID n'{-1) + - -1-lf(bl)D -=-'v( - 1) - 9s9s s s 9s9s s s = 9s9S-1 (f[Ds D~H> + b2 Ds n;<-1>]) = 9s9;- i (f( 4 H)) = 9s'§"S-1< 4 H) 312 (2) (3) - D D'(-l)_,_I + -iv'=<n - l>-i-I - - _ ,_1(4H) 96 6 6 96 96 6 6 96 - 9696 . (4) Putting (1), (2), (3), (4) and (b) together, nn<-1> (~--1)2s ~D=<D -l>-i - _ ,_1(4H) - -'-1(4H) = ~9;9; + ~9; i i 9; +9s9s +9696 17} I (5) (~--1 )2n ~D=<D -I> - _ ,_1(4H) - _,_1(4H) = ~9;9; + ~ ; ; +9s9s +g6g6 · 17} I D is a union of 7 subsets of cosets of H out of 8 possible cosets. Thus, in G / H, the coset representatives used by D fonn an ( 8, 7 , 6) difference set. Other than H, each of the 7 cosets appear 6 times, and each time they are multiplied by 2 H; therefore, each coset is covered 6(2) = 12 times. His covered by :E; D;~-1>, which is the same as the abelian case. This implies D n<-1> = 161 + 12G, (6) so Dis a difference set in G. I Corollary 3.2. The following groups have difference sets as defined in the proof of Theorem 3.1; the groups are defined by their generators, followed by the iso-morphism f and gs, g~, 96, g~ . I . a t6 = 1>4 = I , 2. ba"b-1 =a3 ; bab-1 =iiS; ba"b-1 =a1 ; "ba"b-1 =ii9 ; ba"b-1 =a11 ; 3 . 4. 5. 6 . , 7. ii 16 = b1 = I , ba"b-1 =013; 'ba'b-1 =a1 • a'= b4 ; ba"b-1 =a9 ,a1 =b4 ; "ba"b-1=0-1, a'= b4 ; . f:a4 -+ a4,b2 -+ a4b2 ; 8 . " . 9. Remarks. (I) Not all groups which contain Z4 x Z2 will have difference sets; Z32 x Z2 does not have a dilference set, nor does the group defined by a16 = b4 = 1, bab-1 = a-1• Condition 3.l(a) fails in these cases, and that condition appears to be the most difficult of the three to satisfy. (2) Theorem 3.1 can probably be generalized, but analogies would be needed for Lemma 2.3 and Condition 3. l(a). (3) Groups 3.2.2, 3.2.4, and 3.2.6 are conrained in [3]; they are included here for completeness. 313  References I. J. Davis, Difference sets in abe/ian 2-groups, Ph.D. Dissertation (1987), Uni-versity of Virginia. 2. J. Davis, Difference sets in abe/ian 2-groups. (to appear). 3. J. Davis, Difference sets in nonabe/ian 2-groups, IMA Proceedings (June, 1988). 4. J .F. Dillon, Difference sets in abe/ian 2-groups, Proc. Amer. Math. Soc. (1988). 5. R.G. Kraemer, Proof of a conjecture on Hadamard 2-groups. (to appear). 6. E.S. Lander, Symmetric designs: an algebraic approach, London Math So-ciety Lecture Note Series 74 (1983). 314 

A Note on Nonabelian (64, 28, 12) Difference Sets

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A Note on Nonabelian (64, 28, 12) Difference Sets

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