# 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 .${\ displaystyle (MN) P = M (NP)}$ 