Product of subsets

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 $$A_1A_2 \dots A_n$$ is defined as the set:

$$A_1A_2\dots A_n = \{ a_1a_2\dots a_n \mid a_i \in A_i \}$$

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