## 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?

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?

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?

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

### Re: p-group homomorphism surjective?

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?

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