Editing Semidirect product (section)

== Inner semidirect product definitions ==

Given a group {{math|''G''}} with [[identity element]] {{math|''e''}}, a [[subgroup]] {{math|''H''}}, and a [[normal subgroup]] <math>N \triangleleft G</math>, the following statements are equivalent:
* {{math|''G''}} is the [[product of group subsets#Product of subgroups|product of subgroups]], {{math|1=''G'' = ''NH''}}, and these subgroups have trivial intersection: {{math|1=''N'' ∩ ''H'' = {{mset|''e''}}}}.
* For every {{math|''g'' ∈ ''G''}}, there are unique {{math|''n'' ∈ ''N''}} and {{math|''h'' ∈ ''H''}} such that {{math|1=''g'' = ''nh''}}.
* The [[function composition|composition]] {{math|''π'' ∘ ''i''}} of the natural embedding {{math|''i'' : ''H'' → ''G''}} with the natural projection {{math|''π'' : ''G'' → ''G''/''N''}} induces an [[group isomorphism|isomorphism]] between {{math|''H''}} and the [[quotient group]] {{math|''G''/''N''}}.
* There exists a [[group homomorphism|homomorphism]] {{math|''G'' → ''H''}} that is the [[identity function|identity]] [[restriction (mathematics)|on]] {{math|''H''}} and whose [[kernel (algebra)|kernel]] is {{math|''N''}}. In other words, there is a split [[exact sequence]] <math display="block">1 \to N \to G \to H \to 1</math> of groups (which is also known as a [[Group_extension#Classifying_split_extensions | split extension]] of <math>H</math> by <math>N</math>).
 
If any of these statements holds (and hence all of them hold, by their equivalence), we say {{math|''G''}} is the '''semidirect product''' of {{math|''N''}} and {{math|''H''}}, written
: <math>G = N \rtimes H</math> or <math>G = H \ltimes N,</math>{{efn|The symbol <math>\rtimes</math> is a combination of <math>\triangleleft</math> and <math>\times</math>, oriented so that <math>N \triangleleft (N \rtimes H)</math>.<ref>{{cite web |last1=Neumann |first1=Walter |author-link1=Walter Neumann |title=Notes on semidirect products |page=3 |url=https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/semidirect.pdf#page=3 |access-date=30 December 2024 |archive-url=http://web.archive.org/web/20240716064845/https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/semidirect.pdf#page=3 |archive-date=16 July 2024}}</ref>
}}
or that {{math|''G''}} ''splits'' over {{math|''N''}}; one also says that {{math|''G''}} is a '''semidirect''' product of {{math|''H''}} acting on {{math|''N''}}, or even a semidirect product of {{math|''H''}} and {{math|''N''}}. To avoid ambiguity, it is advisable to specify which is the normal subgroup.

If <math>G = N \rtimes H</math>, then there is a group homomorphism <math>\varphi : H\rightarrow \mathrm{Aut} (N)</math> given by <math>\varphi_h(n)=hnh^{-1}</math>, and for <math>g=nh,g'=n'h'</math>, we have <math>gg'=nhn'h' = nhn'h^{-1}hh' = n\varphi_{h}(n')hh' = n^* h^* </math>.