In group theory, the product of group subsets refers to the set obtained by multiplying each element of one subset by each element of another subset within a given group. Formally, let $G$ be a group with binary operation denoted multiplicatively, and let $A$ and $B$ be subsets of $G$. The product of $A$ and $B$ is the subset
$$ AB = {,ab \mid a \in A,; b \in B,} \subseteq G. $$
Similarly, the product of a finite collection of subsets $A_{1},A_{2},\dots ,A_{n}$ of $G$ is defined iteratively:
$$ A_{1}A_{2}\cdots A_{n}=(((A_{1}A_{2})A_{3})\cdots )A_{n}. $$
When the two subsets coincide, the notation $A^{2}=AA$ is often used.
Basic Properties
- Associativity – Because the group operation in $G$ is associative, the product of subsets is associative: $(AB)C = A(BC)$ for all subsets $A,B,C\subseteq G$.
- Containment of identity – If the identity element $e$ of $G$ belongs to both $A$ and $B$, then $e\in AB$. More generally, if $e\in A$ then $A\subseteq AB$; similarly, if $e\in B$ then $B\subseteq AB$.
- Monotonicity – If $A_{1}\subseteq A_{2}$ and $B_{1}\subseteq B_{2}$ then $A_{1}B_{1}\subseteq A_{2}B_{2}$.
- Inverses – The inverse of a product set satisfies $(AB)^{-1}=B^{-1}A^{-1}$, where $A^{-1}={a^{-1}\mid a\in A}$.
Interaction with Subgroups
- If $H$ and $K$ are subgroups of $G$, their product $HK$ need not be a subgroup. $HK$ is a subgroup if and only if $HK=KH$ (i.e., $H$ and $K$ permute) or equivalently if one of the subgroups normalizes the other.
- When $H$ is a normal subgroup of $G$, the product $HK$ is a subgroup for any subgroup $K\le G$; moreover, $HK$ is the subgroup generated by $H$ and $K$, and the natural map $H\times K\to HK$ is surjective with kernel $H\cap K$.
Applications
- Coset multiplication – For a subgroup $H\le G$ and elements $g_{1},g_{2}\in G$, the product of left cosets satisfies $g_{1}H;g_{2}H=(g_{1}g_{2})H$ if and only if $H$ is normal.
- Double cosets – Given subgroups $H,K\le G$, a double coset is a set of the form $HgK$. The collection of all double cosets partitions $G$.
- Minkowski sum analogue – In additive groups (e.g., $\mathbb{Z}^{n}$ with addition), the product of subsets corresponds to the Minkowski sum $A+B={a+b\mid a\in A,b\in B}$.
Notation Variants
- Some authors denote the product of subsets by $A\cdot B$ or simply juxtaposition $AB$. In additive notation, the operation is written as $A+B$.
- For a single subset $A\subseteq G$, the set of all finite products of elements of $A$ (including repetitions) is denoted $\langle A\rangle$ and constitutes the subgroup generated by $A$.
References
- Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). Wiley. – Section on products of subsets and coset multiplication.
- Rotman, J. J. (1995). An Introduction to the Theory of Groups (4th ed.). Springer. – Discussion of subgroup products and normality.