Re: Many algebra questions

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 08:52:41 +0100

James Pirk wrote:

> I am just going to put this out there to see if there is anything
> someone can help me with. Basically, I am an undergraduate taking a
> graduate algebra class, and everything was going smoothly until the
> last 2 classes. I have no problem working at a highly theoretical
> level, but the professor just assumes just a little too much (I am not
> the only behind, and I hope to help others if I understand).
>
> 1) The first problem is the easiest (in the sense that I actually
> understand it). I'm looking to describe the Sylow subgroups of
> GL(2,Z_5). I know the order: o(GL(2,Z_5))=480=(2^5)*3*5 .
>
> I found some 5-groups, namely matrices of the form ((1 a)(0 1)), where
> (1 a) is the first row, and of course there are 6 of these. Could there
> be more?... I am not sure.

These all must lie in SL(2,Z_5). What's the order of SL(2,Z_5)?
Can it have more than 6 Sylow 5-subgroups?

> Next I found some 8-groups, matrices of the form ((a 0)(b 0)),

?

> and
> there are 15 of these. Not sure if there can be more.

The Sylow 2-subgroups have order 16.

> Finally, I need some 3-groups. These seem to be quite difficult because
> they are going to have nonzero values in all entries. Somehow my
> professor thought the hint H_3\subset Z_5\subset F_25* would help

??

> (where H_3 is a 3-subgroup, F_25 is the field of 25 elements, and F_25*
> is the multiplicative group i.e. nonzero elements).

You have to find matrices of order 3. What are their eigenvalues?

>
> 2) Now start the real problems, most of which I do not understand their
> statement. How many nonisomorphic groups are there with the
> decomposition series "Z_5/Z_7/Z_5 ? Z_7/Z_3/Z_2" . I put that in
> quotations because that is exactly as it appears (with the question
> mark), and I do not understand his notation.

Neither do I.

> 3) Show that S_5=Aut A_5.

FIrst show that S_5 acts as a group of automorphisms of A_5.

> 4) Show that an extension H=A_5\subset G has a section if G/A_5 is
> cyclic group of odd order. Considering I was never told what a
> "section" is, this problem seems hopeless. I could not find that term
> an any literature, on/off-line. Does anyone have a definition?

A homomorphism phi : H -> A_5 with phi(x) = x for x in A_5.

> 5) I know that the rotation group of the cube is S_4 (not sure how to
> prove it though), and from a problem I worked by hand a while ago, I
> know that V_4 and A_4 are the only normal subgroups. So is the
> following:
>
> 1 --> V_4 --> A_4 --> S_4

Odd notation.

> 6) I know, you probably hate me by now. What is the automorphism group
> of Aut(Z_p + Z_{p^2}). Hint: Use the associated filtration of
> Z_p+Z_{p^2}-subgroup of elements of order p. I have never heard my
> professor use the word filtration, so not sure what this means.

What is the p-torsion subgroup of G = Z_p + Z_{p^2}. Call this H.
Then pG is a subset of H. Any automorphism of G induces
automorphisms of H and pG. 

-- 
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_