This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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For a finite set
Let be a finite set. The alternating group on is defined in the following equivalent ways:
- It is the group of all even permutations on under composition. An even permutation is a permutation whose cycle decomposition has an even number of cycles of even size. Specifically, the alternating group on is the subgroup of the symmetric group on comprising the even permutations.
- It is the kernel of the sign homomorphism from the symmetric group on to the group .
For having size zero or one, the alternating group on equals the whole symmetric group on . For having size at least two, the alternating group on is the unique subgroup of index two in the symmetric group on .
The alternating group on a set of size is denoted and is termed the alternating group of degree .
For an infinite set
Let be an infinite set. The finitary alternating group on is defined in the following equivalent ways:
- It is the group of all even permutations on under composition.
- It is the kernel of the sign homomorphism on the finitary symmetric group on .
- The alternating group on a set of size five or more is simple. Also, the finitary alternating group on an infinite set is simple. For full proof, refer: A5 is simple, alternating groups are simple
- Projective special linear group equals alternating group in only finitely many cases
Here, is the degree of the alternating group, i.e., the size of the set it acts on.
|order||for , for||It has index two in the symmetric group of degree .|
|exponent||(skipping over ) if is odd, if is even.|
|nilpotency class||for , undefined for||abelian group for , nilpotent group for .|
|derived length||for , for , undefined for||abelian for , simple for .|
|Frattini length||for , for||Frattini-free group: intersection of maximal subgroups is trivial.|
|Fitting length||for , for , undefined for||abelian for , simple for .|
|minimum size of generating set||2|
|subgroup rank||At most|
|chief length||for , for , for .|
|composition length||for , for , for .|
|Abelian group||Yes for , no for||and don't commute.|
|Nilpotent group||Yes for , no for||is centerless.|
|Solvable group||Yes for , no for||alternating groups are simple for degree five or more.|
|Supersolvable group||Yes for , no for|
|Simple group||Yes for and , no for .|
|Rational group||No for all .|
|Ambivalent group||Yes for , no otherwise||See classification of ambivalent alternating groups.|