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G_{2} F_{4} E_{6} E_{7} E_{8}
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional Lie groups O(∞) SU(∞) Sp(∞)
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_{n} or Alt(n).
Contents
Basic properties
For n > 1, the group A_{n} is the commutator subgroup of the symmetric group S_{n} with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : S_{n} → {1, −1} explained under symmetric group.
The group A_{n} is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A_{5} is the smallest nonabelian simple group, having order 60, and the smallest nonsolvable group.
The group A_{4} has a Klein fourgroup V as a proper normal subgroup, namely the double transpositions {(12)(34), (13)(24), (14)(23)}, and maps to A_{3} = C_{3}, form 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.
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