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alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set is called the alternating group of degree ''n'', or the alternating group on ''n'' letters and denoted by ''A''''n'' or Alt(''n'').
== Basic properties ==
For , 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 explained under symmetric group.
The group ''A''''n'' is abelian if and only if and simple if and only if or . ''A''5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.
The group ''A''4 has a Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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