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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 if 1 = g ∈ G and g / ∈ S, then g can be written as the product ab−1, 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. 1 2 3 4 5 6 7 8 9 10 11 12

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 A (v, k, λ, µ)-PDS is called regular if 1 / ∈ S and S = S−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. 1 2 3 4 5 6 7 8 9 10 11 12

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 vO(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 p3 and exponent p2. We have the following presentation for G: G = x, y, z | xp2 = yp = zp = 1, xp = z, [x, z] = [y, z] = 1, [y, x] = z. Theorem There exists a regular (p3, p2 + p − 2, p − 2, p + 2)-PDS S of G. Corollary If S is the regular (p3, p2 + p − 2, p − 2, p + 2)-PDS from above, then the Cayley graph Cay(G, S) is a (p3, p2 + 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 | xp2 = yp = zp = 1, xp = z, [x, z] = [y, z] = 1, [y, x] = z. Let j be a generator of (Z/p2Z)×, so that p(p − 1) is the smallest value of n for which jn = 1 in Z/p2Z. We define automorphisms σ and φ of G as follows: σ :    x → xyz

p+1 2

y → y z → z φ :    x → xjp y → y z → zj

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 Zp:Zp−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 | xp2 = yp = zp = 1, xp = z, [x, z] = [y, z] = 1, [y, x] = z. Define: S1 := xH, S2 := y\{1}, S3 := zH, and S := S1 ∪ S2 ∪ S3.

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 | xp2 = yp = zp = 1, xp = z, [x, z] = [y, z] = 1, [y, x] = z. S = xH ∪ y\{1} ∪ zH. SH = S, xH has p(p − 1) = |H| elements, y\{1} has p − 1 elements, zH 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

Introduction Definitions

Checking that S is really a regular PDS

G = x, y, z | xp2 = yp = zp = 1, xp = z, [x, z] = [y, z] = 1, [y, x] = z. S = xH ∪ y\{1} ∪ zH For the g in S, it suffices to check the number of ways that x, z, and powers of y can be written as products ab−1, a, b ∈ S. For the g not in S, since every conjugacy class meets S, g = ch for some c ∈ S, h ∈ G. As above, it suffices to check exactly one c from each orbit of H. The conjugates of elements of zH are in the center

• f G, so it suffices to check conjugates of x and conjugates of powers
• f y.

Computation is a little tedious, but not terrible, and everything works! S is a regular (p3, p2 + p − 2, p − 2, p + 2)-PDS of G.

Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 18 / 20

Introduction Definitions

Future directions

Tip of the iceberg. Other families of p-groups? What about other groups??? Could interesting new partial geometries (or even GQs) be constructed from these methods?

Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 19 / 20

Introduction Definitions

Thanks!

Eric Swartz (The University of Western Australia) New examples of partial difference sets in finite nonabelian groups 5 August 2013 20 / 20