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.