MathDB

For a group

\left( G, \star \right)

and

A, B

two non-void subsets of

G

, we define

A \star B = \left\{ a \star b : a \in A \ \text{and}\ b \in B \right\}

. (a) Prove that if

n \in \mathbb N, \, n \geq 3

, then the group \left( \mathbb Z \slash n \mathbb Z,+\right) can be writen as \mathbb Z \slash n \mathbb Z = A+B, where

A, B

are two non-void subsets of \mathbb Z \slash n \mathbb Z and A \neq \mathbb Z \slash n \mathbb Z, \, B \neq \mathbb Z \slash n \mathbb Z, \, \left| A \cap B \right| = 1. (b) If

\left( G, \star \right)

is a finite group,

A, B

are two subsets of

G

and

a \in G \setminus \left( A \star B \right)

, then prove that function

f : A \to G \setminus B

given by

f(x) = x^{-1}\star a

is well-defined and injective. Deduce that if

|A|+|B| > |G|

, then

G = A \star B

. [hide="Question."]Does the last result have a name?

Problem Statement