Complex product

The complex product , usually simply called product , is a term from a mathematical sub-area , group theory .

Is a group and are and subsets of , then the complex product of with is defined as ${\displaystyle (G,\cdot )}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle N}$

${\displaystyle M\cdot N:=\{m\cdot n\mid m\in M,n\in N\}}$.

There are also the abbreviations

${\displaystyle {\begin{aligned}MN&:=M\cdot N\\gN&:=\{g\}\cdot N\\Mg&:=M\cdot \{g\}\end{aligned}}}$

common, where is an element of the group. ${\displaystyle g}$

Since the above definition only requires the existence of a two-digit link , the complex product can also be viewed in more general structures, for example in semigroups .

properties

• The complex product of two subgroups and is a union of left subclasses of and a union of right subclasses of : ${\displaystyle UV}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle U}$

${\displaystyle UV=\bigcup _{u\in U}uV=\bigcup _{v\in V}Uv}$

• If and are finite subgroups, then the equation holds for the thickness of the complex product ${\displaystyle U}$ ${\displaystyle V}$

${\displaystyle |UV|={\frac {|U|\cdot |V|}{|U\cap V|}}}$

• The complex product of a normal subgroup with a subgroup gives a subgroup. More precisely, it holds for all subgroups and that is a subgroup if and only if applies. If or is a normal divisor, this is fulfilled. In particular, the complex product of subgroups in Abelian groups is a subgroup. ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle UV}$ ${\displaystyle UV=VU}$ ${\displaystyle U}$ ${\displaystyle V}$
• The complex product of secondary classes and a normal divisor is . With this product, the secondary classes of normal factors form a group, the factor group of after . ${\displaystyle gN}$ ${\displaystyle hN}$ ${\displaystyle N}$ ${\displaystyle gN\cdot hN=(gh)N}$ ${\displaystyle G}$ ${\displaystyle N}$
• Is the normal divisor and subgroup of that have the properties and , then the inner semidirect product of with . For the existence of such a subgroup for a given normal subgroup, reference is made to the Schur-Zassenhaus theorem . ${\displaystyle N}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle N\cap U=\{e\}}$ ${\displaystyle N\cdot U=G}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle U}$
• The power set of a group, together with the complex product no group, yet still Monoid , in particular the complex product is associative , so . ${\displaystyle (MN)P=M(NP)}$

literature

• Thomas W. Hungerford: Algebra . 5. print. Springer-Verlag, 1989, ISBN 0-387-90518-9