Product of group subsets

In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a group G then their product is the subset of G defined by

ST=\{st:s\in S{\text{ and }}t\in T\}

Note that S and T need not be subgroups. The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of G.

If S and T are subgroups of G their product need not be a subgroup (consider, for example, two distinct subgroups of order two in S₃). It will be a subgroup if and only if ST = TS and the two subgroups are said to permute. In this case ST is the group generated by S and T, i.e. ST = TS = <S ∪ T>. If either S or T is normal then this condition is satisfied and ST is a subgroup. Suppose S is normal. Then according to the second isomorphism theorem S ∩ T is normal in T and ST/S ≅ T/(S ∩ T).

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size |ST| given by the product formula:

|ST|={\frac {|S||T|}{|S\cap T|}}

Note that this applies even if neither S nor T is normal.

In particular, if S and T (subgroups now) intersect only in the identity, then every element of ST has a unique expression as a product st with s in S and t in T. If S and T also commute, then ST is a group, and is called a Zappa–Szep product. Even further, if S or T is normal in ST, then ST is called a semidirect product. Finally, if both S and T are normal in ST, then ST is called a direct product.

References

Rotman, Joseph (1995). An Introduction to the Theory of Groups (4th ed.). Springer-Verlag. ISBN 0-387-94285-8.

See also

References