alternating group has index 2 in the symmetric group, the

We prove that the alternating groupMathworldPlanetmath An has index 2 in the symmetric groupMathworldPlanetmathPlanetmath Sn, i.e., An has the same cardinality as its complement SnAn. The proof is function-theoretic. Its idea is similar to the proof in the parent topic, but the focus is less on algebraic aspect.

Let πSnAn. Define π:SnAnAn by π(σ)=πσ, where πσ is the productPlanetmathPlanetmath of π and σ.



since π-1 exists and π-1πσ=π-1πδ.

Onto: Given αAn, there exists an element in SnAn, namely λ=π-1α, such that


(The element λ is in SnAn because π-1 is and the product of an odd permutationMathworldPlanetmath and an even permutation is odd.)

The function π:SnAnAn is, therefore, a one-to-one correspondence, so both sets SnAn and An have the same cardinality.

Title alternating group has index 2 in the symmetric group, the
Canonical name AlternatingGroupHasIndex2InTheSymmetricGroupThe
Date of creation 2013-03-22 16:48:49
Last modified on 2013-03-22 16:48:49
Owner yesitis (13730)
Last modified by yesitis (13730)
Numerical id 8
Author yesitis (13730)
Entry type Proof
Classification msc 20-00