Suppose D is a (upsilon, k, lambda)-difference set over the integers modulo upsi

Question

Suppose D is a (upsilon, k, lambda)-difference set over the integers modulo upsilon. (i) If m is an integer prime to upsilon, define mD to be the set formed by multiplying the members of D by m and reducing modulo upsilon. Prove that mD is a (upsilon, k, lambda)-difference set. (ii) If m is prime to upsilon, and mD equals D or some shift of D, then m is called a multiplier of D. Prove that the set of multipliers of D form a group under multiplication modulo upsilon. (iii) Prove that the condition "m is prime to upsilon" in part (i) is necessary.

Explanation / Answer

Let D = {a,b,c,d....}

Consider any element a in D

am mod v elements will form a set as

{am mod v, bm mod v, cm modv,.....}

As D is a (V,k,lemda) set the set formed by {am mod v, bm mod v, cm modv,.....} is also a (V,k,lemda) set

---------------------------

Let mD = D or shift in D

Then mD = {p+a, p+b,.....} for some p ranging from 0 to m-1

Since D is a group it is closed under addition and have inverse

Since a+d = c is in D

p+a+p+b = p+p+c will be in mD hence closure is true and inverse of p+a will be p+a-1

Hence group.

---------------------

If m is not prime to v, say gcd (m,v) = q then

multiplying by m will not have the difference set as

m = qr hence mD will be mqr and will not have the same difference.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.