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:$ST\; =\; \{st\; :\; s\; in\; S\; mbox\{\; and\; \}\; tin\; T\}$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. 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" &cup; "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" &cap; "T" is normal in "T" and "ST"/"S" &cong; "T"/("S" &cap; "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"::$|ST|\; =\; frac\{|Scap\; T$Note that this applies even if neither "S" nor "T" is normal.

In particular, if "S" and "T" 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 permute, 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.

ee also

*direct product (group theory)
*semidirect product

References

*cite book
first = Joseph
last = Rotman
year = 1995
title = An Introduction to the Theory of Groups
edition = (4th ed.)
publisher = Springer-Verlag
id = ISBN 0-387-94285-8