# What is Difference Set of a Group

By Xah Lee. Date:

A difference set of a group is a subset of the group that satisfies some properties.

Let there be a group (G,⊕). The difference set of the group is a subset of G, such that, for any non-identity element of G, it can be expressed as α⊕β^-1 in exactly λ ways, where α and β are elements of D.

A difference set is said to be (v,k,λ) difference set, where v is the size of G, k is the size of D, and λ.

## Example of a Difference Set

suppose we have ℤ13, the cyclic group of order 13. That is, with elements 0 to 12, and modular arithmetic as operation.

the set D = {0,1,3,9} is a (13,4,1) difference set of the group ℤ13.

first, let's get a table of ℤ13's inverse elements.

(Reminder: two elements {x,y} are inverse to each other if x⊕y=y⊕x=0, where 0 is the identity element. (a identity element is a element (let's denote it “0”) such that z⊕0=0⊕z=z for any element z.))

Let's denote the inverse of a element x as inv[x].

• inv = 0
• inv = 12
• inv = 11
• inv = 10
• inv = 9
• inv = 8
• inv = 7
• inv = 6
• inv = 5
• inv = 4
• inv = 3
• inv = 2
• inv = 1

Here is a table, showing that every non-identity element of ℤ13, is expressed as α⊕inv[β] in 1 way, where α and β are in D. (and it's the only way)

• 1 = 1 ⊕ inv = 1 ⊕ 0
• 2 = 3 ⊕ inv = 3 ⊕ 12
• 3 = 3 ⊕ inv = 3 ⊕ 0
• 4 = 0 ⊕ inv = 0 ⊕ 4
• 5 = 1 ⊕ inv = 1 ⊕ 4
• 6 = 9 ⊕ inv = 9 ⊕ 10
• 7 = 3 ⊕ inv = 3 ⊕ 4
• 8 = 9 ⊕ inv = 9 ⊕ 12
• 9 = 9 ⊕ inv = 9 ⊕ 0
• 10 = 0 ⊕ inv = 0 ⊕ 10
• 11 = 1 ⊕ inv = 1 ⊕ 10
• 12 = 0 ⊕ inv = 0 ⊕ 12

Note: it's called “difference set” because, if we denote group operator by “+”, then, we might also denote inverse of β by `-β` by convention, thus `α + inv[β]` is written as `α + -β` or just `α - β`, thus “difference”.

## Reference

the example is from Bill Cherowitzo http://www-math.ucdenver.edu/~wcherowi/'s file at http://www-math.ucdenver.edu/~wcherowi/courses/m6406/diffsets.pdf