Re: Sylow subgroups

in article <11kkk4jc78mkta5@xxxxxxxxxxxxxxxxxx>,
jim heckman <wnzrfeurpxzna@xxxxxxxxxxxxxxxxx> wrote:

|On 9-Oct-2005, jdolan@xxxxxxxxxxxxxxxxxxxxxxxx (James Dolan) wrote in
|message <dib5c7$l5k$1@xxxxxxxxxxxx>:

|> in article <11khrhfivmut861@xxxxxxxxxxxxxxxxxx>,
|> jim heckman <wnzrfeurpxzna@xxxxxxxxxxxxxxxxx> wrote:
|>
|> |On 8-Oct-2005, mark12345 <enphinion@xxxxxxxxxxx> wrote in message
|> |<15867057.1128799313005.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>:
|> |
|> |> Thank you very much, this was very helpful, however, I have trouble
|> |> seeing why is PGL(2,5) isomorphic to S_5?
|> |
|> |Since PGL(2,5) has index 6 in S_6, it must be isomorphic to the point
|> |stabilizer of the latter in its natural action.
|>
|> assuming that i understand what you're saying, isn't that a funny way
|> of putting it? it seems like the real point of what you're saying is
|> that pgl(2,5) as a subgroup of 6! must be _conjugate_ to the point
|> stablizer of 6!
|
|No, I meant the isomorphism comes from the action of S_6 on the set
|of cosets of G = PGL(2,5). Derek Holt correctly divined my thinking
|in his reply to you. G and S_5 are clearly *not* conjugate in S_6,
|which is to say they're not permutation isomorphic.

that was my point: that your point was that g is conjugate _not_ to
the stablizer in the natural action, but rather to something else:
thus to the stablizer in some "unnatural" action; a fact which is
quite worth pointing out (because it's confusing not to, but moreso,
because it gives an elegant way of seeing that 6!, unlike any other
symmetric group, has such an unnatural action; that is, that it has a
non-trivial outer automorphism.)

derek holt missed the point as well. the moment that you give an
invertible homomorphism h from the permutations of one 6-element set s
to the permutations of another 6-element set s', with h clearly not
induced by any bijection from s to s', you're showing (whether you
like it or not) that 6! has a non-trivial outer automorphism.

anyway, i have to admit that that was an amusing place for you to put
a mid-sentence break in your quoting of my post.




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