Editing Group extension (section)

====Classifying split extensions====

A '''split extension''' is an extension

:<math>1\to K\to G\to H\to 1</math>

with a [[homomorphism]] <math>s\colon H \to G</math> such that going from ''H'' to ''G'' by ''s'' and then back to ''H'' by the quotient map of the short exact sequence induces the [[identity function|identity map]] on ''H'' i.e., <math>\pi \circ s=\mathrm{id}_H</math>.  In this situation, it is usually said that ''s'' '''splits''' the above [[exact sequence]].

Split extensions are very easy to classify, because an extension is split [[if and only if]] the group ''G'' is a [[semidirect product]] of ''K'' and ''H''. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from <math>H\to\operatorname{Aut}(K)</math>, where Aut(''K'') is the [[automorphism]] group of ''K''.  For a full discussion of why this is true, see [[semidirect product]].