New necessary conditions for Payley type PDS in Abelian groups

TL;DR

This paper establishes new necessary conditions on the prime factorization of the order of Abelian groups that can contain Paley type partial difference sets, specifically linking prime congruences to exponents.

Contribution

It provides novel necessary conditions on the prime factors of group order for the existence of Paley type PDS in Abelian groups, extending previous results.

Findings

If a regular Paley type PDS exists, each prime with odd exponent must be congruent to 3 mod 4.

These conditions restrict the possible orders of Abelian groups admitting such PDS.

The results are analogous to known conditions for Abelian Hadamard difference sets.

Abstract

In this paper we prove that if there is a regular Paley type partial difference set in an Abelian group $G$ of order $v$, where $v=p_1^{2k_1}p_2^{2k_2}\cdots p_n^{2k_n}$, $n\ge 2$, $p_1$, $p_2$, $\cdots$, $p_n$ are distinct odd prime numbers, then for any $1 \le i \le n$, $p_i$ is congruent to 3 modulo 4 whenever $k_i$ is odd. These new necessary conditions further limit the specific order of an Abelian group $G$ in which there can exist a Paley type partial difference set. Our result is similar to a result on Abelian Hadamard (Menon) difference sets proved by Ray-Chaudhuri and Xiang in 1997.

Equations

$$\left(p^{2s+1},\frac{p^{2s+1}-1}{2},\frac{p^{2s+1}-5}{4},\frac{p^{2s+1}-1}{4}\right),$$

$$v_1=|N|,{\beta }_1={\lambda }_1-{\mu }_1=\beta -2\theta \pi ,{\Delta }_1={\beta }_1^2+4(k_1-{\mu }_1)={\pi }^2$$

$$k_1=\frac{1}{2}\left[|N|+{\beta }_1\pm \sqrt{(|N|+{\beta }_1{)}^2-({\Delta }_1-{\beta }_1^2)(|N|-1)}\right].$$

$$(nh{)}^{x+t{p}^r}=n{h}^x\in D.$$

$$\mu=\mu_{1}+s(p_{i}-1)=\frac{|N|-1}{4}+s(p_{i}-1)\;\;\mbox{ for some integer $s$}.$$

$$\mu -{\mu }_1=s(p_i-1)=\frac{|G|-|N|}{4}=p_1^{2k_1}{p}_2^{2k_2}\cdots p_{i-1}^{2k_{i-1}}{p}_{i+1}^{2k_{i+1}}\cdots p_n^{2k_n}\frac{p_i^{2k_i}-1}{4}.$$

$$p_i^{2k_i}-1=(p_i-1)(p_i^{2k_i-1}+p_i^{2k_i-2}+\cdots +p_i+1),$$

$$p_i-1|p_1^{2k_1}{p}_2^{2k_2}\cdots p_{i-1}^{2k_{i-1}}{p}_{i+1}^{2k_{i+1}}\cdots p_n^{2k_n}\frac{p_i^{2k_i}-1}{4}$$

$$p_i^{2k_i-1}+p_i^{2k_i-2}+\cdots +p_i+1\equiv 0(\mathrm{mod}4).$$

Full text

1 Introduction and the main result

Let $G$ be a finite Abelian group of order $v$, and let $D\subseteq G$ be a subset of size $k$. We say $D$ is a $(v,k,\lambda,\mu)$-partial difference set (PDS) in $G$ if the expressions $gh^{-1}$, $g$, $h\in D$, $g\neq h$, represent each non-identity element in $D$ exactly $\lambda$ times, and each non-identity element of $G$ not in $D$ exactly $\mu$ times. If we further assume that $D^{(-1)}=D$ (where $D^{(s)}=\{g^{s}:g\in D\}$) and $e\notin D$ (where $e$ is the identity element of $G$), then $D$ is called a regular partial difference set. A regular PDS is called trivial if $D\cup\{e\}$ or $G\setminus D$ is a subgroup of $G$. The condition that $D$ be regular is not a very restrictive one, as $D^{(-1)}=D$ is automatically fulfilled whenever $\lambda\neq\mu$, and $D$ is a PDS if and only if $D\cup\{e\}$ is a PDS. The Cayley graph over $G$ with connection set $D$, denoted by Cay($G$, $D$), is the graph with the elements of $G$ as vertices, and in which two vertices $g$ and $h$ are adjacent if and only if $gh^{-1}$ belongs to $D$. When the connection set $D$ is a regular partial difference set, Cay($G$, $D$) is a strongly regular graph. The importance of regular PDSs lies in the fact that they are equivalent to strongly regular Cayley graphs. For more information on partial difference sets, we refer the reader to a survey of Ma [8].

Throughout this paper, we will use the following standard notations: $\beta=\lambda-\mu$ and $\Delta=\beta^{2}+4(k-\mu)$.

Partial difference sets with parameters $(v,(v-1)/2,(v-5)/4,(v-1)/4)$ are called Paley type partial difference sets. Over the last three decades this subject has seen active research, see for example [1], [2], [4] (Theorem 3.2), [5], [9], [11]. There are two key problems on Paley type PDSs in Abelian groups:

For what order of the group, can we find Paley-type PDSs?

• 2.

In which type of groups of given order, can we find Paley-type PDSs?

The second question seems largely out of reach , although there is some literature on this topic, see for example [2], [5]. In this paper, we will focus on question 1. It is well-known that when $q\equiv 1\pmod{4}$ and $q$ is a prime power, the non-zero squares of a finite field ${\mathbb{F}}_{q}$ form a Paley type PDS in the additive group of $\mathbb{F}_{q}$. A further important result was proved by S.L. Ma in 1984:

Theorem 1.1

[6]* Let $D$ be an Abelian regular $(v,k,\lambda,\mu)$-PDS, and assume that $\Delta$ is not a perfect square. Then $D$ is of Paley type; more precisely, $D$ has parameters*

[TABLE]

where $p$ is a prime congrent to 1 modulo 4.

Let $D$ be a regular Paley type PDS in an Abelian group $G$, where $|G|=v$. Then $\Delta=(-1)^{2}+4(\frac{v-1}{4})=v$. If $v$ is not a square, by Theorem 1.1, $|G|=v=p^{2s+1}$ for some prime $p\equiv 1\pmod{4}$. For a prime power $q\equiv 1\pmod{4}$, we can always construct a Paley type PDS in $(\mathbb{F}_{q},+)$ using the non-zero squares of the finite field $\mathbb{F}_{q}$. Thus to answer question 1, we only need to focus on the existence of Paley type PDSs when $v$ is a perfect square and not a prime power, that is, when $v=p_{1}^{2k_{1}}p_{2}^{2k_{2}}\cdots p_{n}^{2k_{n}}$, $n\geq 2$.

When $|G|=p_{1}^{2k_{1}}p_{2}^{2k_{2}}\cdots p_{n}^{2k_{n}}$, $p_{1}$, $p_{2}$, $\cdots$, $p_{n}$ are distinct odd prime numbers, and all $k_{i}$s are even, Polhill ([9]) constructed Paley type PDSs in $G=\mathbb{Z}_{p_{1}}^{2k_{1}}\times\mathbb{Z}_{p_{2}}^{2k_{2}}\times\cdots\times\mathbb{Z}_{p_{n}}^{2k_{n}}$. In this paper we focus on which of the $k_{i}$s would possibly be odd.

We now state our main theorem.

Theorem 1.2

Let $G$ be an Abelian group of order $v$, where $v=p_{1}^{2k_{1}}p_{2}^{2k_{2}}\cdots p_{n}^{2k_{n}}$, $n\geq 2$, $p_{1}$, $p_{2}$, $\cdots$, $p_{n}$ are distinct odd prime numbers. If there is a regular Paley type PDS in $G$, then for any $1\leq i\leq n$, if $k_{i}$ is odd, we have $p_{i}\equiv 3\pmod{4}$.

At this point, we want to point out a strikingly similar result on Abelian Hadamard (Menon) difference sets proved by Ray-Chaudhuri and Xiang in 1997.

When $\lambda=\mu$, then a $(v,k,\lambda,\mu)$ partial difference set is called a $(v,k,\lambda)$ difference set (DS). Hadamard (Menon) difference sets, having parameters $(4m^{2},2m^{2}-m,m^{2}-m)$, are of particular interest due to their connections with Hadamard matrices.

Theorem 1.3

[10]* If there is a Hadamard difference set in an Abelian group $G=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times P$, where $|P|=p^{2\alpha}$, $\alpha$ is odd, $p$ is an odd prime number, then $p$ is a prime congruent to 3 modulo 4.*

Although we expect there should be a deeper reason explaining this similarity, we are currently not aware of any general argument.

2 Proof of the Main Result

Below we cite three results on Abelian regular partial difference sets. The first of these was proved by S.L. Ma in [7], the second one was proved by K.T. Arasu, D. Jungnickel, S.L. Ma and A. Pott in [1], and the last one, the local multiplier theorem, was proved by S. De Winter, E. Kamischke, and Z. Wang in [3].

Proposition 2.1

[7]* Let D be a nontrivial regular $(v,\,k,\,\lambda,\,\mu)$-PDS in an Abelian group G. Suppose $\Delta$ is a perfect square. If $N$ is a subgroup of $G$ such that $\gcd(\left|N\right|,\left|G\right|/\left|N\right|)=1$ and $\left|G\right|/\left|N\right|$ is odd, then $D_{1}=D\cap N$ is a (not necessarily non-trivial) regular $(v_{1},\,k_{1},\,\lambda_{1},\,\mu_{1})$-PDS with*

[TABLE]

and

[TABLE]

where $\pi=\gcd(|N|,\sqrt{\Delta})$ and $\theta$ is the integer satisfying $(2\theta-1)\pi\leq\beta<(2\theta+1)\pi$.

Theorem 2.2

[1]* Let $\Gamma$ be a strongly regular Cayley graph based on an Abelian group $G$, with parameters $v$, $k$, $\lambda$, and $\mu$ satisfying $\beta=\lambda-\mu=-1$. Then, up to complementation, $\Gamma$ is either of Paley type or it has parameters $(243,22,1,2)$.*

Theorem 2.3

[3]* Let D be a regular $(v,k,\lambda,\mu)$-PDS in an Abelian group $G$. Furthermore assume $\Delta$ is a perfect square. Then $g\in G$ belongs to $D$ if and only if $g^{s}$ belongs to $D$ for all $s$ coprime with $o(g)$, the order of $g$.*

Using Theorem 2.2, we can now prove the following:

Lemma 2.4

Let $D$ be a regular Paley type PDS in an Abelian group $G$, where $|G|=v$ is a perfect square. If $N$ is a non-trivial subgroup of $G$ such that $\gcd(\left|N\right|,\left|G\right|/\left|N\right|)=1$, $\left|G\right|/\left|N\right|$ is odd, and $|N|\neq 243$, then $D_{1}=D\cap N$ is a regular Paley type PDS in $N$.

Proof: Clearly $\Delta=(\lambda-\mu)^{2}+4(k-\mu)=1+(v-1)=v$. Applying Proposition 2.1, we have $(2\theta-1)\pi\leq\beta=-1<(2\theta+1)\pi$. Since $\pi=\gcd(|N|,\sqrt{\Delta})\geq 1$, it follows that $\theta=0$. Hence $\beta_{1}=\beta-2\theta\pi=-1$. By Theorem 2.2, $D_{1}=D\cap N$ is a regular Paley type PDS in $N$.$\Box$

Lemma 2.5

Let $G=N\times H$ be an Abelian group, where $N$ and $H$ are subgroups of order $q$ and $p^{k}$ respectively. Also, assume that $\gcd(p,q)=1$, $p$ is an odd prime number, and $q$ is a positive odd integer. Let $g=nh\in D$ with $n\in N$ and $h\in H\setminus\{1_{H}\}$, $o(h)=p^{r}$. Then $nh\in D$ if and only if $nh^{x}\in D$ for all $x$ satisfying $1\leq x\leq p^{r}-1$, and $\gcd(x,p)=1$.

Proof: Let $x$ be a positive integer satisfying $1\leq x\leq p^{r}-1$ and $\gcd(x,p)=1$. Since $\gcd(p,q)=1$, it follows that $x$, $x+p^{r}$, $\cdots$, $x+(q-1)p^{r}$ are in different residue classes modulo $q$. Thus there exists an integer $t$, $0\leq t\leq q-1$, such that $x+tp^{r}\equiv 1\pmod{q}$. Clearly, $\gcd(x+tp^{r},q)=1$ and $\gcd(x+tp^{r},p^{r})=\gcd(x,p^{r})=1$. Since $\gcd(p,q)=1$, it follows that $\gcd(x+tp^{r},p^{r}q)=1$. As $g=nh\in D$ and ${\rm o}(g)\,|\,p^{r}q$, by Theorem 2.3 (the Local Multiplier Theorem), we have

[TABLE]

$\Box$

Now we are ready to prove our main theorem.

Proof: Here $v=p_{1}^{2k_{1}}p_{2}^{2k_{2}}\cdots p_{n}^{2k_{n}}$, and it is easy to check that $\Delta=v$.

For any $i$, $1\leq i\leq n$, we let $G=N\times H$, where $N$ and $H$ are subgroups of $G$ with orders $v/p_{i}^{2k_{i}}$ and $p_{i}^{2k_{i}}$ repectively. Next we assume that $D$ is a regular Paley type PDS in $G$. As $|N|\neq 3^{5}=243$, by Lemma 2.4, $D_{1}=D\cap N$ is a Paley type PDS in $N$ with $\mu_{1}=\frac{|N|-1}{4}$. Let $n$ be any non-identity element of $N$ and $n\notin D_{1}$. In total, there are exactly two mutually exclusive types of representations of $n$ as differences from $D$, and these are as follows:

• (i)

there are $\mu_{1}$ representations of $n$ of the form $n_{1}n_{2}^{-1}$ with $n_{1},\,n_{2}\in D_{1}$;

• (ii)

$n$ can also be written as $(n_{1}h)(n_{2}h)^{-1}$ if $n=n_{1}n_{2}^{-1}$, $n_{1},\,n_{2}\in N$, and $n_{1}h\in D$, $n_{2}h\in D$, $h\in H\setminus\{1_{H}\}$.

By Lemma 2.5, if $n_{1}h,\,n_{2}h\in D$, so are $n_{1}h^{x},\,n_{2}h^{x}$ for any $x$ with $1\leq x\leq{\rm o}(h)-1$ and $\gcd(x,{\rm o}(h))=1$. Thus the second type of representations always appear in a set of size $\phi(o(h))$, where $o(h)=p_{i}^{r}$ for some $r$ with $1\leq r\leq 2k_{i}$. As $\phi(o(h))=p_{i}^{r}-p_{i}^{r-1}$ is always divisible by $p_{i}-1$, we have

[TABLE]

On the other hand, $\mu=\frac{|G|-1}{4}$. It follows that

[TABLE]

It follows that $p_{i}-1\,|\,p_{1}^{2k_{1}}p_{2}^{2k_{2}}\cdots p_{i-1}^{2k_{i-1}}p_{i+1}^{2k_{i+1}}\cdots p_{n}^{2k_{n}}\frac{p_{i}^{2k_{i}}-1}{4}$.

Since

[TABLE]

and $p_{1}$, $p_{2}$, $\cdots$, $p_{n}$ are odd prime numbers,

[TABLE]

holds only when

[TABLE]

From Congruence (1), it easily follows that if $k_{i}$ is odd, we have $p_{i}\equiv 3\pmod{4}$. This proves the theorem. $\Box$

**Sample Application: ** By Theorem 1.2, there does not exist a regular (225, 112, 55, 56)-PDS in Abelian groups since $225=3^{2}\times 5^{2}$ and $5\equiv 1\pmod{4}$. But there are strongly regular graphs with parameters (225, 112, 55, 56). More generally, there does not exist regular Paley type PDSs of order $25q^{2}$ in Abelian groups, where $\gcd(5,q)=1$.

Given our main result a first natural question is: Does there exist a regular Paley type PDS in an Abelian group of order $v=p^{2k}q^{2}$, where $\gcd(p,q)=1$, $p$ is a prime number congruent to 3 modulo 4, and $k$ is odd? For example, does there exist a regular Paley type PDS of order $3^{2}7^{2}$, or a regular Paley type PDS of order $3^{4}7^{2}$?