Corollary

*The homomorphism h defined in 2) of Proposition 27 is injective if and only if f and g are injective and $f(N) \cap g(L) = \{e\}$; and h is surjective if and only if $f(N)g(L) = G$.

Since $h(x, y) = f(x)g(y)$, the second assertion is obvious; moreover it follows from (4) that $f(N)g(L) = g(L)f(N)$, and the first assertion follows from the remarks at the beginning of this sub-section.

The group S defined in Proposition 27 is said to be the external semi-direct product of N and L (relative to $\sigma$); we shall generally identify N (resp. L) with the normal subgroup $j_1(N)$ [resp. the subgroup $j_2(L)$] of S. If $\sigma_y$ is the identity element of $\Gamma$ for all $y \in L$, we retrieve the usual notion of the product of two groups.

Now let G be a group, and let L and N be two subgroups of G such that LN = NL and such that N is normal in NL, so that for each $y \in L$, $\sigma_y : x \to yxy^{-1}$ is an automorphism of N, and $y \to \sigma_y$ is a homomorphism of L into the automorphism group $\Gamma$ of N. It follows then from Proposition 27 that if S is the external semi-direct product of N and L (relative to $\sigma$), then $h : (x, y) \to xy$ is a homomorphism of S into G. h is bijective if and only if we have $N \cap L = \{e\}$ and $NL = G$ (Corollary to Proposition 27); G is then said to be the semi-direct product of its normal subgroup N and its subgroup L, and we often identify G with S by means of h.