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 s'3). 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 = . 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 &cap; t is normal in t and st/s &cong; t/(s &cap; 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 zappaâszÃ©p 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. ==see also== *direct product of groups *semidirect product ==references== *
Binary_operations Group_theory