Editing Semidirect product (section)

===Outer semidirect product===
Let us now consider the outer semidirect product. Given any two groups {{math|''N''}} and {{math|''H''}} and a group homomorphism {{math|''φ'' : ''H'' → Aut(''N'')}}, we can construct a new group {{math|''N'' ⋊{{sub|''φ''}} ''H''}}, called the '''outer semidirect product''' of {{math|''N''}} and {{math|''H''}} with respect to {{math|''φ''}}, defined as follows:<ref>{{cite book |last1=Robinson |first1=Derek John Scott |title=An Introduction to Abstract Algebra |year=2003 |publisher=[[De Gruyter|Walter de Gruyter]] |isbn=9783110175448 |pages=75–76}}</ref>
{{numbered list
| The underlying set is the [[Cartesian product]] {{math|''N'' × ''H''}}.
| The group operation <math>\bullet</math> is determined by the homomorphism {{math|''φ''}}:
: <math>\begin{align}
  \bullet : (N \rtimes_\varphi H) \times (N \rtimes_\varphi H) &\to N \rtimes_\varphi H\\
                                         (n_1, h_1) \bullet (n_2, h_2) &= (n_1 \varphi_{h_1}(n_2),\, h_1 h_2)
\end{align}</math>
for {{math|''n''<sub>1</sub>, ''n''<sub>2</sub>}} in {{math|''N''}} and {{math|''h''<sub>1</sub>, ''h''<sub>2</sub>}} in {{math|''H''}}.
}}

This defines a group in which the identity element is {{math|(''e''<sub>''N''</sub>, ''e<sub>H</sub>'')}} and the inverse of the element {{math|(''n'', ''h'')}} is {{math|(''φ''<sub>''h''<sup>−1</sup></sub>(''n''<sup>−1</sup>), ''h''<sup>−1</sup>)}}. Pairs {{math|(''n'', ''e<sub>H</sub>'')}} form a normal subgroup isomorphic to {{math|''N''}}, while pairs {{math|(''e<sub>N</sub>'', ''h'')}} form a subgroup isomorphic to {{math|''H''}}. The full group is a semidirect product of those two subgroups in the sense given earlier.

Conversely, suppose that we are given a group {{math|''G''}} with a normal subgroup {{math|''N''}} and a subgroup {{math|''H''}}, such that every element {{math|''g''}} of {{math|''G''}} may be written uniquely in the form {{math|''g {{=}} nh''}} where {{math|''n''}} lies in {{math|''N''}} and {{math|''h''}} lies in {{math|''H''}}. Let {{math|''φ'' : ''H'' → Aut(''N'')}} be the homomorphism (written {{math|''φ''(''h'') {{=}} ''φ''<sub>''h''</sub>}}) given by
: <math>\varphi_h(n) = hnh^{-1}</math>
for all {{math|''n'' ∈ ''N'', ''h'' ∈ ''H''}}.

Then {{math|''G''}} is isomorphic to the semidirect product {{math|''N'' ⋊{{sub|''φ''}} ''H''}}. The isomorphism {{math|''λ'' : ''G'' → ''N'' ⋊{{sub|''φ''}} ''H''}} is well defined
by {{math|''λ''(''a'') {{=}} ''λ''(''nh'') {{=}} (''n, h'')}} due to the uniqueness of the decomposition {{math|''a'' {{=}} ''nh''}}.

In {{math|''G''}}, we have 
: <math>(n_1 h_1)(n_2 h_2) = n_1 h_1 n_2(h_1^{-1}h_1) h_2 =
    (n_1 \varphi_{h_1}(n_2))(h_1 h_2)</math>
Thus, for {{math|''a'' {{=}} ''n''{{sub|1}}''h''{{sub|1}}}} and {{math|''b'' {{=}} ''n''{{sub|2}}''h''{{sub|2}}}} we obtain
: <math>\begin{align}
\lambda(ab) & = \lambda(n_1 h_1 n_2 h_2) = \lambda(n_1 \varphi_{h_1} (n_2) h_1 h_2) = (n_1 \varphi_{h_1} (n_2), h_1 h_2) = (n_1, h_1) \bullet (n_2, h_2) \\[5pt]
& = \lambda(n_1 h_1) \bullet \lambda(n_2 h_2) = \lambda(a) \bullet \lambda(b),
\end{align}</math>
which [[mathematical proof|proves]] that {{math|''λ''}} is a homomorphism. Since {{math|''λ''}} is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in {{math|''N'' ⋊{{sub|''φ''}} ''H''}}.

The direct product is a special case of the semidirect product. To see this, let {{math|''φ''}} be the trivial homomorphism (i.e., sending every element of {{math|''H''}} to the identity automorphism of {{math|''N''}}) then {{math|''N'' ⋊{{sub|''φ''}} ''H''}} is the direct product {{math|''N'' × ''H''}}.

A version of the [[splitting lemma]] for groups states that a group {{math|''G''}} is isomorphic to a semidirect product of the two groups {{math|''N''}} and {{math|''H''}} [[if and only if]] there exists a [[Exact sequence#Short exact sequence|short exact sequence]]
: <math> 1 \longrightarrow N \,\overset{\beta}{\longrightarrow}\, G \,\overset{\alpha}{\longrightarrow}\, H \longrightarrow 1</math>
and a group homomorphism {{math|''γ'' : ''H'' → ''G''}} such that {{math|''α''&thinsp;∘&thinsp;''γ'' {{=}} id<sub>''H''</sub>}}, the identity map on {{math|''H''}}. In this case, {{math|''φ'' : ''H'' → Aut(''N'')}} is given by {{math|''φ''(''h'') {{=}} ''φ''<sub>''h''</sub>}}, where
:<math>\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).</math>