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(By "useful" I mean, "useful to prove other theorems".)

I understand Cayley's theorem ("every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$"), but I don't see what one can do with it.

Granted, there is great merit in the unifying view that it gives of all groups, but I'd like to know of concrete deductions and lines of proof that Cayley's theorem makes possible.

In other words, I'd love to see examples where Cayley's theorem "saves the day" (i.e. makes an otherwise difficult/intractable proof easy/tractable).

kjo
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    The order of the subgroup must divide the order of the group. So If you know the order of the group ... – mick Jan 31 '18 at 01:42
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    Cayley graphs can be used to prove or disprove the possibility to embed a group $G$ in some symmetric group $S_m$. I am not completely sure, but they might provide a simpler answer to this question, for instance. – Jack D'Aurizio Jan 31 '18 at 01:49
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    Suppose you want to prove that every finite group is embeddable in a $2$-generated finite group. There are other ways to prove it, but is any of them as easy as the proof using permutations? Likewise if you want to prove that every countable group is embeddable in a $2$-generated group; I think the easiest proof is the one using permutations. – bof Jan 31 '18 at 01:51

1 Answers1

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Here is a nice application of Cayley Theorem:

Lemma Every group with $4n+2$ elements contains a subgroup of index $2$.

Proof: By Cayley Theorem $G$ is isomorphich with a subgroup $H$ of $S_{4n+2}$, and every $\sigma \in H, \sigma \neq e$ has no fixed points.

Next, as every group of even order, there exists some $\sigma \in H$ of order $2$. Since $\sigma$ has order $2$, it the product of transpositions. Also, since $\sigma$ has no fixed point, it is the product of $2n+1$ transpositions.

Therefore $\sigma$ is an odd permutation.

Then $\tau \to \sigma \tau$ is a bijection between the set of even and odd permutations in $H$.

It follows that $H \cap A_n$ has index $2$ in $H$.

As a consequence we get:

Corollary For $n \geq 1$, group with $4n+2$ elements is not simple.

Try to prove directly that a group with $4n+2$ elements cannot be simple for $n \geq 1$.

Or try to prove without Cayley Theorem the Lemma stated above.

N. S.
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