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 pgroups 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?
pgroup homomorphism surjective?
Re: pgroup homomorphism surjective?
This is exactly why I am an applied mathematician...

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 Joined: Tue Aug 09, 2011 6:18 pm
Re: pgroup 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 pgroups 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
owlpride, if you've already figured solved the problem, please post your solution. I'd enjoy seeing it
Re: pgroup 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 pgroups:
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 pgroup B, then A is a proper subgroup of its normalizer in P.)
2. The other special property of pgroups is that they have nontrivial center, which sometimes allows for induction on the number of group elements. (Mod out by the center, get a pgroup of lower order, apply inductive hypothesis and show that you can go back up.)
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 pgroup B, then A is a proper subgroup of its normalizer in P.)
2. The other special property of pgroups is that they have nontrivial center, which sometimes allows for induction on the number of group elements. (Mod out by the center, get a pgroup of lower order, apply inductive hypothesis and show that you can go back up.)
Re: pgroup homomorphism surjective?
I know a student who works on pgroups, i could ask him in a week if i remember.