p-group homomorphism surjective?

Forum for the GRE subject test in mathematics.
owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

p-group homomorphism surjective?

Postby owlpride » Sat Mar 03, 2012 1:06 am

This is not a MGRE question but I'm stuck and I thought that this might be good practice for you as well.

Suppose G and H are p-groups and f: G -> H is a group homomorphism, and suppose that the induced map G/[G,G] -> H/[H,H] is surjective. Show that f is surjective.

Any ideas how to approach this?

ackirby
Posts: 5
Joined: Mon Oct 31, 2011 9:52 pm

Re: p-group homomorphism surjective?

Postby ackirby » Sun Mar 04, 2012 5:17 pm

This is exactly why I am an applied mathematician... :?

Topoltergeist
Posts: 44
Joined: Tue Aug 09, 2011 6:18 pm

Re: p-group homomorphism surjective?

Postby Topoltergeist » Mon Mar 05, 2012 6:55 pm

I think this is a pretty neat question. Wikipedia says that f([G,G]) is a subgroup of [H,H]. If we can show that f([G,G]) = [H,H], then it should follow that f is surjective. But I don't see how p-groups are going to help ... ... ... is it possible to use the Sylow theorems?

owlpride, if you've already figured solved the problem, please post your solution. I'd enjoy seeing it :D

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

Re: p-group homomorphism surjective?

Postby owlpride » Mon Mar 05, 2012 8:20 pm

I do not yet have a solution (though there are several people who claim to know how to do this, but are too busy to tell me...) I can think of two ways to use that we have p-groups:

1. If we could show that the normalizer of f(G) in H is all of H, then f(G) = H. (If A is a proper subgroup of a p-group B, then A is a proper subgroup of its normalizer in P.)

2. The other special property of p-groups is that they have non-trivial center, which sometimes allows for induction on the number of group elements. (Mod out by the center, get a p-group of lower order, apply inductive hypothesis and show that you can go back up.)

PNT
Posts: 37
Joined: Fri Mar 11, 2011 9:01 pm

Re: p-group homomorphism surjective?

Postby PNT » Mon Mar 12, 2012 4:18 am

I know a student who works on p-groups, i could ask him in a week if i remember.




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