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In [[mathematics]], one can define a '''product of group subsets''' in a natural way. If ''S'' and ''T'' are [[subset]]s of a [[group (mathematics)|group]] ''G'' then their product is the subset of ''G'' defined by | |||
:<math>ST = \{st : s \in S \text{ and } t\in T\}</math> | |||
Note that ''S'' and ''T'' need not be [[subgroup]]s. 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''<sub>''3''</sub>). It will be a subgroup if and only if ''ST'' = ''TS'' and the two subgroups are said to [[permutable subgroup|permute]]. In this case ''ST'' is the group [[generating set of a group|generated]] by ''S'' and ''T'', i.e. ''ST'' = ''TS'' = <''S'' ∪ ''T''>. If either ''S'' or ''T'' is [[normal subgroup|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'': | |||
:<math>|ST| = \frac{|S||T|}{|S\cap T|}</math> | |||
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 of groups|direct product]]. | |||
==See also== | |||
*[[Direct product of groups]] | |||
*[[Semidirect product]] | |||
==References== | |||
*{{cite book | |||
| first = Joseph | |||
| last = Rotman | |||
| year = 1995 | |||
| title = An Introduction to the Theory of Groups | |||
| edition = 4th | |||
| publisher = Springer-Verlag | |||
| isbn = 0-387-94285-8 | |||
}} | |||
[[Category:Group theory]] | |||
[[Category:Binary operations]] |
Latest revision as of 11:13, 9 October 2013
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
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:
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.
See also
References
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