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

$ST\; =\; \backslash \{st\; :\; s\; \backslash in\; S\; \backslash mbox\{\; and\; \}\; t\backslash in\; T\backslash \}$

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. It will be a subgroup if and only if ST = TS. In this case ST is the group generated by S and T, i.e. ST = TS = <S ∪ T>. If either S or T is a 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 and subgroups of G then the order of ST is given by the product formula:

$|ST|\; =\; \backslash frac\{|S||T|\}\{|S\backslash cap\; T|\}$

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

See also

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• direct product (group theory)
• semidirect product

References

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Group theory | Binary operations

Produkt von Gruppen