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' ^-1kh'k' , and h' ^-1kh is an element of K, so that hh' is in H and h' ^-1kh'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.