New examples of partial difference sets in finite nonabelian groups Eric Swartz The University of Western Australia 5 August 2013
Introduction Definitions Definition The groups, graphs, etc., considered in this talk will be finite. Definition A subset S of elements of a group G is a ( v , k , λ, µ )- partial difference set (PDS) if | G | = v , | S | = k , if 1 � = g ∈ G and g ∈ S , then g can be written as the product ab − 1 , where a , b ∈ S , exactly λ different ways, and ∈ S , then g can be written as the product ab − 1 , if 1 � = g ∈ G and g / where a , b ∈ S , exactly µ different ways. Why partial difference set? Originally interest was in abelian groups, and the operation was addition. Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 2 / 20
Introduction Definitions Small example Example Let G be the additive group of GF (13), and let S = { 1 , 3 , 4 , 9 , 10 , 12 } . For elements in S : 1 = 4 − 3 = 10 − 9 3 = 4 − 1 = 12 − 9 4 = 3 − 12 = 1 − 10 9 = 12 − 3 = 10 − 1 10 = 1 − 4 = 9 − 12 12 = 3 − 4 = 9 − 10 Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 3 / 20
Introduction Definitions Small example Example Let G be the additive group of GF (13), and let S = { 1 , 3 , 4 , 9 , 10 , 12 } . For nonidentity elements not in S : 2 = 3 − 1 = 12 − 10 = 1 − 12 5 = 9 − 4 = 1 − 9 = 4 − 12 6 = 9 − 3 = 10 − 4 = 3 − 10 7 = 3 − 9 = 4 − 10 = 10 − 3 8 = 4 − 9 = 9 − 1 = 12 − 4 11 = 1 − 3 = 10 − 12 = 12 − 1 Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 3 / 20
Introduction Definitions Small example Example Let G be the additive group of GF (13), and let S = { 1 , 3 , 4 , 9 , 10 , 12 } . For nonidentity elements not in S : 2 = 3 − 1 = 12 − 10 = 1 − 12 5 = 9 − 4 = 1 − 9 = 4 − 12 6 = 9 − 3 = 10 − 4 = 3 − 10 7 = 3 − 9 = 4 − 10 = 10 − 3 8 = 4 − 9 = 9 − 1 = 12 − 4 11 = 1 − 3 = 10 − 12 = 12 − 1 S is a (13 , 6 , 2 , 3)-PDS. Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 3 / 20
Introduction Definitions Paley’s Theorem In fact, the last example generalizes: Theorem (Paley 1933) Let G be the additive group of a finite field GF ( q ) , where q is an odd prime power and q ≡ 1 (mod 4) . Then the set S of all nonzero squares in GF ( q ) forms a ( q , q − 1 2 , q − 5 4 , q − 1 4 ) -PDS in G. Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 4 / 20
Introduction Definitions Cayley graphs Definition The Cayley graph Cay ( G , D ) is defined to be the graph whose vertex set is the elements of G such that g , h ∈ G are adjacent if and only if gh − 1 ∈ D . Note that Cay ( G , D ) is an undirected graph if and only if D = D − 1 , i.e., for each d ∈ D , d − 1 ∈ D . Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 5 / 20
Introduction Definitions Example Example Let G again be the additive group of of GF (13) and let S = { 1 , 3 , 4 , 9 , 10 , 12 } . Cay ( G , S ) is an undirected Cayley graph. 3 4 5 2 1 6 0 7 12 8 11 9 10 Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 6 / 20
Introduction Definitions Strongly regular graphs and regular PDSs Definition A graph Γ is a ( v , k , λ, µ )- strongly regular graph if Γ has v vertices, Γ is regular of degree k , any two adjacent vertices are mutually adjacent to exactly λ other vertices, and any two nonadjacent vertices are mutually adjacent to exactly µ other vertices. Definition ∈ S and S = S − 1 . A ( v , k , λ, µ )-PDS is called regular if 1 / Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 7 / 20
Introduction Definitions Example Example Let G again be the additive group of of GF (13) and let S = { 1 , 3 , 4 , 9 , 10 , 12 } . S is a regular (13 , 6 , 2 , 3)-PDS and Cay ( G , S ) is an undirected (13 , 6 , 2 , 3)-strongly regular Cayley graph. 3 4 5 2 1 6 0 7 12 8 11 9 10 Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 8 / 20
Introduction Definitions Equivalent concepts Proposition Let G be a finite group and let S be a regular ( v , k , λ, µ ) -PDS. Then Cay ( G , S ) is a ( v , k , λ, µ ) -strongly regular graph. EXERCISE: Prove it! Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 9 / 20
Introduction Definitions Why are they interesting? Cayley graphs are useful in applications, such as the construction of expander graphs . Partial geometries such as finite generalized quadrangles have a related point graph that is a strongly regular graph. Very few known for nonabelian groups! Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 10 / 20
Introduction Definitions A heuristic approach, useful in practice Let’s suppose that we have a regular ( v , k , λ, µ )-PDS S in a finite group G . Cay ( G , S ) is a ( v , k , λ, µ )-strongly regular Cayley graph. Many known examples (coming from finite GQs, for instance) have additional automorphisms, other than G , acting on the graph! Pick a group of “outer automorphisms” H of G and assume that S is invariant under H . Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 11 / 20
Introduction Definitions Pros and cons Advantages Cuts down search time tremendously. Linear programming is now feasible. Invariance of S under automorphisms makes proving that it works much easier. Big disadvantage No guarantee that there are many extra automorphisms! As v goes to infinity, strongly regular Cayley graphs with v vertices have v O (log( v )) automorphisms. See recent work by Babai, Wilmes (2013) and Chen, Sun, Teng (2013). Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 12 / 20
Introduction Definitions Main result Let p be an odd prime and let G be the extraspecial group of order p 3 and exponent p 2 . We have the following presentation for G : G = � x , y , z | x p 2 = y p = z p = 1 , x p = z , [ x , z ] = [ y , z ] = 1 , [ y , x ] = z � . Theorem There exists a regular ( p 3 , p 2 + p − 2 , p − 2 , p + 2) -PDS S of G. Corollary If S is the regular ( p 3 , p 2 + p − 2 , p − 2 , p + 2) -PDS from above, then the Cayley graph Cay ( G , S ) is a ( p 3 , p 2 + p − 2 , p − 2 , p + 2) -strongly regular graph. Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 13 / 20
Introduction Definitions Outer automorphisms of G G = � x , y , z | x p 2 = y p = z p = 1 , x p = z , [ x , z ] = [ y , z ] = 1 , [ y , x ] = z � . Let j be a generator of ( Z / p 2 Z ) × , so that p ( p − 1) is the smallest value of n for which j n = 1 in Z / p 2 Z . We define automorphisms σ and φ of G as follows: p +1 x �→ xyz 2 σ : y �→ y z �→ z x j p x �→ φ : y �→ y z j z �→ Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 14 / 20
Introduction Definitions Properties of σ and φ Define H := � σ, φ � ≤ Aut ( G ) . Lemma If σ and φ are the automorphisms defined above, then: (i) The order of σ is p; (ii) The order of φ is (p-1); (iii) The group H defined above is isomorphic to Z p : Z p − 1 , the Frobenius group of order p ( p − 1) . Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 15 / 20
Introduction Definitions Constructing the PDS G = � x , y , z | x p 2 = y p = z p = 1 , x p = z , [ x , z ] = [ y , z ] = 1 , [ y , x ] = z � . Define: S 1 := x H , S 2 := � y �\{ 1 } , S 3 := z H , and S := S 1 ∪ S 2 ∪ S 3 . Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 16 / 20
Introduction Definitions Properties of S G = � x , y , z | x p 2 = y p = z p = 1 , x p = z , [ x , z ] = [ y , z ] = 1 , [ y , x ] = z � . S = x H ∪ � y �\{ 1 } ∪ z H . S H = S , x H has p ( p − 1) = | H | elements, � y �\{ 1 } has p − 1 elements, z H has p − 1 elements, S = S − 1 and 1 / ∈ S , For each nonidentity conjugacy class C of G , |C ∩ S | = 1 . Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 17 / 20
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