Product of subsets

Definition

Terminology

For an abelian group, the notion of product of subsets defined here is also termed the Minkowski sum.

For two subsets

Suppose $G$ is a group and $A,B$ are (possibly equal, possibly distinct) subsets of $G$. The product of subsets $AB$ is defined as the set:

$AB=\{ab\mid a\in A,b\in B\}$

Note that if $G$ is non-abelian, then $AB$ may differ from $BA$.

For finitely many subsets

Suppose $G$ is a group and $A_{1},A_{2},\dots ,A_{n}$ are (possibly equal, possibly distinct) subsets of $G$. The product of subsets $A_{1}A_{2}\dots A_{n}$ is defined as the set:

$A_{1}A_{2}\dots A_{n}=\{a_{1}a_{2}\dots a_{n}\mid a_{i}\in A_{i}\}$

Facts

Lower bounds on size

• Product of subsets whose total size exceeds size of group equals whole group
• Cauchy-Davenport theorem: A lower bound for a group of prime order.
• Kemperman's theorem: A lower bound on the measure for compact connected groups.

Upper bounds on size

• Size of product of subsets is bounded by product of sizes