Basic Properties
For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → {1, −1} explained under symmetric group.
The group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.
The group A4 has a Klein four-group V as a proper normal subgroup, namely the double transpositions {(12)(34), (13)(24), (14)(23)}, and maps to, from the sequence In Galois theory, this map, or rather the corresponding map 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.
Read more about this topic: Alternating Group
Famous quotes containing the words basic and/or properties:
“The gay world that flourished in the half-century between 1890 and the beginning of the Second World War, a highly visible, remarkably complex, and continually changing gay male world, took shape in New York City.... It is not supposed to have existed.”
—George Chauncey, U.S. educator, author. Gay New York: Gender, Urban Culture, and the Making of the Gay Male World, 1890-1940, p. 1, Basic Books (1994)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)