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Cayley's theorem (Theorem)

Let $ G$ be a group, then $ G$ is isomorphic to a subgroup of the permutation group $ S_{G}$

If $ G$ is finite and of order $ n$, then $ G$ is isomorphic to a subgroup of the permutation group $ S_{n}$

Furthermore, suppose $ H$ is a proper subgroup of $ G$. Let $ X = \{Hg \vert g \in G\}$ be the set of right cosets in $ G$. The map $ \theta:G \to S_{X}$ given by $ \theta(x)(Hg) = Hgx$ is a homomorphism. The kernel is the largest normal subgroup of $ H$. We note that $ \vert S_X\vert = [G : H]!$. Consequently if $ \vert G\vert$ doesn't divide $ [G : H]!$ then $ \theta$ is not an isomorphism so $ H$ contains a non-trivial normal subgroup, namely the kernel of $ \theta$.



"Cayley's theorem" is owned by vitriol.
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proof of Cayley's theorem (Proof) by Evandar
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Cross-references: contains, isomorphism, divide, normal subgroup, kernel, homomorphism, map, right cosets, proper subgroup, order, finite, permutation group, subgroup, isomorphic, group
There is 1 reference to this entry.

This is version 4 of Cayley's theorem, born on 2002-02-19, modified 2002-04-14.
Object id is 2174, canonical name is CayleysTheorem.
Accessed 6292 times total.

Classification:
AMS MSC: 20B35 (Group theory and generalizations :: Permutation groups :: Subgroups of symmetric groups)
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