Re: Groups of order 30, 105 and pqr

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 10/11/04 

Date: Mon, 11 Oct 2004 13:32:08 +0100

Sylvan Jacques wrote:

> I am working some problems in Milne's excellent new version
> (8/29/2003) of his text on Group Theory for 1st year grad students.
>
> Prob 64. Prove or give counter-example:
> (a) Every group of order 30 has a normal subgroup of order 15.
> (b) Every group of order 30 is nilpotent.
> --------------
>
> The Sylow thms --> either n_5 or n_3 = 1, i.e., either the subgroup
> N such that |N| = 5 or H such that |H| = 3 is normal.
>
> This works if |G| = 105 = 3.5.7, and I think for |G| = pqr, product of 3
> primes.

Would it work if |G| = 465 = 3.5.31?

> If n_5 = 1, and N = <x>, |G/N| = 6 --> there is a y' in G' = G/N such that
> |y'| = 3,
>
> and if H' = <y'>, H' is normal in G'.

OK

> Then there is y in G such that y' = yN = y <x>, so there is
>
> z = yx in G, and |y| = 3, |x| = 5, so |z| = 15.
>
> I want to say that M = <z> is normal in G, which seems to be true,
> but its not clear to me how to prove it.

It is true that yN is finite, but all you need is that yN is
a subgroup. It is so since it's the inverse image of H'
under the projection from G to G/N.

What if n_5 = 6? 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

Relevant Pages

  • Re: Proving primality of Mersenne primes
    ... > Is there a general method for proving primality of primes of the form ... Yes: the Lucas-Lehmer test. ... Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html ...
    (sci.math)
  • Re: Groups of order 30, 105 and pqr
    ... Sylvan Jacques wrote: ... >> a subgroup. ... It is so since it's the inverse image of H' ... Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html ...
    (sci.math)
Quantcast