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From: Dmitry Sustretov on 20 Jan 2010 07:35
Hi,

I am trying to find an *elementary* solution to the following problem:
prove that S_n is not isomorphic to a direct product of two non-
trivial groups. Now, one can come up with a solution that uses the
simplicity of A_n, but I have encountered this problem in a context
where the reader is not supposed to even know what a normal group is.

So, the question is how one can prove this fact using just the
definitions of direct product and symmetric group.

--
Dmitry Sustretov
From: Tonico on 20 Jan 2010 08:19
On Jan 20, 2:35 pm, Dmitry Sustretov <dmitry.sustre...(a)gmail.com>
wrote:
> Hi,
>
> I am trying to find an *elementary* solution to the following problem:
> prove that S_n is not isomorphic to a direct product of two non-
> trivial groups. Now, one can come up with a solution that uses the
> simplicity of A_n, but I have encountered this problem in a context
> where the reader is not supposed to even know what a normal group is.


This doesn't seem to make much sense: direct and semi-direct products
involve normal group(s)
intersecting in the trivial subgroup...Not to mention that dealing
with direct products without first learning about normal subgroups
sounds like trying to understand integration without first knowing
derivatives.

Could you please provide a reference to this problem where the student
isn't supposed to know about normal subgroups?

Tonio


>
> So, the question is how one can prove this fact using just the
> definitions of direct  product and symmetric group.
>
> --
> Dmitry Sustretov

From: Timothy Murphy on 20 Jan 2010 09:02
Tonico wrote:

> I am trying to find an *elementary* solution to the following problem:
>> prove that S_n is not isomorphic to a direct product of two non-
>> trivial groups. Now, one can come up with a solution that uses the
>> simplicity of A_n, but I have encountered this problem in a context
>> where the reader is not supposed to even know what a normal group is.
>
>
> This doesn't seem to make much sense: direct and semi-direct products
> involve normal group(s)
> intersecting in the trivial subgroup...Not to mention that dealing
> with direct products without first learning about normal subgroups
> sounds like trying to understand integration without first knowing
> derivatives.

>> So, the question is how one can prove this fact using just the
>> definitions of direct product and symmetric group.

I would have thought one could prove this easily enough
simply from the fact that if S_n = G x H
then every element of G must commute with every element of H.
After all, there are not that many elements commuting with g in S_n
unless g = e.

I don't have space to give the complete proof here ...

--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: Dmitry Sustretov on 20 Jan 2010 09:21
The reference wouldn't be of much use --- it is in Russian. It is a
course in algebra designed for high-school students, and the problem
appears right after the definition of the direct product.

I am fairly sure that an elementary solution is possible. This is just
sheer curiosity, of course. Perhaps someone can see the solution quite
easily; personally, I am not very good at such "brute force",
combinatorial arguments.

> > Hi,
>
> > I am trying to find an *elementary* solution to the following problem:
> > prove that S_n is not isomorphic to a direct product of two non-
> > trivial groups. Now, one can come up with a solution that uses the
> > simplicity of A_n, but I have encountered this problem in a context
> > where the reader is not supposed to even know what a normal group is.
>
> This doesn't seem to make much sense: direct and semi-direct products
> involve normal group(s)
> intersecting in the trivial subgroup...Not to mention that dealing
> with direct products without first learning about normal subgroups
> sounds like trying to understand integration without first knowing
> derivatives.
>
> Could you please provide a reference to this problem where the student
> isn't supposed to know about normal subgroups?
>
> Tonio
>
>
>
>
>
> > So, the question is how one can prove this fact using just the
> > definitions of direct  product and symmetric group.
>
> > --
> > Dmitry Sustretov

From: Derek Holt on 20 Jan 2010 10:30
On 20 Jan, 12:35, Dmitry Sustretov <dmitry.sustre...(a)gmail.com> wrote:
> Hi,
>
> I am trying to find an *elementary* solution to the following problem:
> prove that S_n is not isomorphic to a direct product of two non-
> trivial groups. Now, one can come up with a solution that uses the
> simplicity of A_n, but I have encountered this problem in a context
> where the reader is not supposed to even know what a normal group is.
>
> So, the question is how one can prove this fact using just the
> definitions of direct  product and symmetric group.

Well here is one possible approach - I don't know whether this is
sufficiently elementary.

Assume S_n = G x H with both nontrivial.

Using Bertrand's postulate, choose a prime p with n/2 < p <= n.

Since p divides n! but p^2 does not, one of the factors G and H, say
G, contains an element g of order p,
which must be a single p-cycle, say (1,2,3,...,p).

Now comes the bit that may be too hard. Any element h in the
centralizer of g in S_n must act as a power of g on {1,2,...,p} and
must permute the remaining points {p+1,...,n} among themselves. It is
certainly possible to prove that using completely elementary
arguments. This applies to every element of H, and since |H| is not
divisible by p, every element of H must fix all of 1,2,...,p.

So the fixed point set X of H is nonempty and is a proper subset of
{1,2,...,n}, since H is nontrivial. But every element of G is in the
centralizer of H, and hence must permute the points of X among
themselves. So every element of G and of H and hence every element of
S_n stabilizes X, which is false.

Derek Holt.

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