Editing Alternating group (section)

== Basic properties ==
For {{nowrap|''n'' > 1}}, the group A<sub>''n''</sub> is the [[commutator subgroup]] of the [[symmetric group]] S<sub>''n''</sub> with [[Index of a subgroup|index]] 2 and has therefore [[factorial|''n''!]]/2 elements. It is the [[kernel (algebra)|kernel]] of the signature [[group homomorphism]] {{nowrap|sgn : S<sub>''n''</sub> → {{mset|1, −1}}}} explained under [[symmetric group]].

The group A<sub>''n''</sub> is [[abelian group|abelian]] [[if and only if]] {{nowrap|''n'' ≤ 3}} and [[simple group|simple]] if and only if {{nowrap|1=''n'' = 3}} or {{nowrap|''n'' ≥ 5}}.<!-- Note A3 is in fact a simple group of order 3. A1 and A2 are groups of order 1, so not usually called simple, and A4 has a non-identity proper normal subgroup so is not simple. -->  A<sub>5</sub> is the smallest non-abelian [[simple group]], having [[order of a group|order]] 60, and thus the smallest non-[[solvable group]].

The group A<sub>4</sub> has the [[Klein four-group]] V as a proper [[normal subgroup]], namely the identity and the double transpositions {{nowrap|{{mset| (),  (12)(34), (13)(24), (14)(23) }}}}, that is the kernel of the [[surjection]] of A<sub>4</sub> onto {{nowrap|1=A<sub>3</sub> ≅ Z<sub>3</sub>}}. We have the [[exact sequence]] {{nowrap|1=V → A<sub>4</sub> → A<sub>3</sub> = Z<sub>3</sub>}}. In [[Galois theory]], this map, or rather the corresponding map {{nowrap|S<sub>4</sub> → S<sub>3</sub>}}, corresponds to associating the [[Lagrange resolvent]] cubic to a quartic, which allows the [[quartic polynomial]] to be solved by radicals, as established by [[Lodovico Ferrari]].