# Product of groups

In

mathematics, given a

group *G* and two

subgroups *H* and

*K* of

*G*, one can define the

**product** of

*H'\' and *K

*, denoted by *HK'' as the set of all
elements of the form

*hk*, for all

*h* in

*H* and

*k* in

*K*. In
general

*HK* is not a subgroup (

*hkh'k' * is not of the form

*hk*);
it is a subgroup if and only if one among

*H* and

*K* is a

normal subgroup of

*G*. Indeed, if this is the case (assume

*K* is normal),

*hkh'k' * =

*hh' h' *^{-1}*kh'k' *, and

*h' *^{-1}*kh* is
an element of

*K*, so that

*hh' * is in

*H* and

*h' *^{-1}*kh'k' * is in

*K*, as required. An analogous argument shows that (

*hk*)

^{-1} is of the form

*h'k' *.

Of particular interest are products enjoying further properties, the semidirect product and the direct product. They allow also to construct a product of two groups not given as subgroups of a fixed group.