alternating group is a normal subgroup of the symmetric group
Theorem 1.
The alternating group An is a normal subgroup of the symmetric group Sn
Proof.
Define the epimorphism f:Sn→ℤ2 by :σ↦0 if σ is an even permutation and :σ↦1 if σ is an odd permutation. Hence, An is the kernel of f and so it is a normal subgroup of the domain Sn. Furthermore Sn/An≅ℤ2 by the first isomorphism theorem. So by Lagrange’s theorem
|Sn|=|An||Sn/An|. |
Therefore, |An|=n!/2. That is, there are n!/2 many elements in An ∎
Remark. What we have shown in the theorem is that, in fact, An has index 2 in Sn. In general, if a subgroup H of G has index 2, then H is normal in G. (Since [G:H]=2, there is an element g∈G-H, so that gH∩H=∅ and thus gH=Hg).
Title | alternating group is a normal subgroup of the symmetric group |
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Canonical name | AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup |
Date of creation | 2013-03-22 13:42:32 |
Last modified on | 2013-03-22 13:42:32 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Theorem |
Classification | msc 20-00 |