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alternating group is a normal subgroup of the symmetric group (Theorem)
Theorem 1   The alternating group $ A_{n}$ is a normal subgroup of the symmetric group $ S_{n}$
Proof. Define the epimorphism $ f:S_{n}\rightarrow\mathbb{Z}_2$ by $ :\sigma\mapsto 0$ if $ \sigma$ is an even permutation and $ :\sigma\mapsto 1$ if $ \sigma$ is an odd permutation. Hence, $ A_{n}$ is the kernel of $ f$ and so it is a normal subgroup of the domain $ S_{n}$. Furthermore $ S_{n}/A_{n}\cong\mathbb{Z}_2$ by the first isomorphism theorem. So by Lagrange's theorem
$\displaystyle \vert S_{n} \vert=\vert A_{n} \vert\vert S_{n}/A_{n}\vert. $
Therefore, $ \vert A_{n}\vert=n!/2$. That is, there are $ n!/2$ many elements in $ A_{n}$ $ \qedsymbol$

Remark. What we have shown in the theorem is that, in fact, $ A_n$ has index $ 2$ in $ S_n$. 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\in G-H$, so that $ gH\cap H=\varnothing$ and thus $ gH=Hg$).



"alternating group is a normal subgroup of the symmetric group" is owned by CWoo. [ full author list (3) | owner history (2) ]
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the alternating group has index 2 in the symmetric group (Proof) by yesitis
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Cross-references: subgroup, index, Lagrange's theorem, first isomorphism theorem, domain, kernel, odd permutation, even permutation, epimorphism, symmetric group, normal subgroup, alternating group

This is version 5 of alternating group is a normal subgroup of the symmetric group, born on 2003-06-23, modified 2007-08-04.
Object id is 4387, canonical name is AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup.
Accessed 4345 times total.

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AMS MSC: 20-00 (Group theory and generalizations :: General reference works )
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