Math Is Fun Forum

mcbrike

Member

Registered: 2010-08-20

Posts: 2

group theory (simple groups)

G has order pqr (p > q> r).  It's easy to use Sylow Theorems directly to show G is not simple (just a counting argument based on n(p), n(q), n(r) -- the numbers of Sylow p-, q-, and r-subgroups).

But, can we say more in this case. E.g., I would suspect that the Syl p-group is unique. Perhaps the Syl q-gp as well.

Any ideas?

Jim

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: group theory (simple groups)

The number of Sylow p-subgroups is of the form 1+kp (k an integer) and divides qr. The only such number is 1. Hence the Sylow p-subgroup is unique. Similarly the Sylow q- and r-subgroups are also unique.

You might find these threads in the Euler Avenue section useful:

http://www.mathisfunforum.com/viewtopic.php?id=10734
http://www.mathisfunforum.com/viewtopic.php?id=12125

mcbrike

Member

Registered: 2010-08-20

Posts: 2

Re: group theory (simple groups)

Jane,

Wish it were that easy. Take G of order 7*5*3.  n(7) = 15 is both 1 (mod 7) and divides 5*3.  And n(5) could be 21; and n(3) could be 7.

So there is no guaranteed uniquess of the Sylow subgp for any of the prime factors.

Any other ideas?

Jim

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: group theory (simple groups)

Sorry, my mistake. The Sylow p-subgroup would be unique if p > qr (which was what I was probably thinking of), otherwise it need not be. The other Sylow subgroups also need not be unique.

Last edited by JaneFairfax (2010-08-20 07:10:46)