[r-t] Group thy. again

Martin Bright martin at boojum.org.uk
Fri Feb 15 10:05:26 UTC 2013

On 15 February 2013 11:40, Robin Woolley <robin at robinw.org.uk> wrote:
> Today's question generalises this. Given any group Sn, is An unique of all
> the subgroups of size n!/2? (i.e., the only one.)

Yes.  For n > 4, this follows from the non-trivial but well-known fact
that An is simple, i.e. has no non-trivial proper normal subgroups.
Any subgroup of index 2 in Sn has to be normal, so must intersect An
in a normal subgroup, so must be An.  For n=2 and n=3 it's easy, and
n=4 needs a little calculation (A4 isn't simple, but its only
non-trivial proper normal subgroup is of index 3.)

There may also be a more elementary argument.


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